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File: Calculus Pdf 168928 | Calculus Cheat Sheet Integrals Reduced
calculus cheat sheet calculus cheat sheet integrals standard integration techniques definitions note that at many schools all but the substitution rule tend to be taught in a calculus ii class ...

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                                                                                                 Calculus Cheat Sheet                                                                                                                                                                                                                      Calculus Cheat Sheet 
                                                                                                                 Integrals                                                                                                                                                                                                       Standard Integration Techniques 
                                                                                                                 Definitions                                                                                                                                Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class. 
                  Definite Integral: Suppose  fx is continuous                                                                   Anti-Derivative : An anti-derivative of  fx                                                                                 
                                                                                   ( )                                                                                                                                      ( )                                                                                                                                                         bgb
                                                                                                                                                                                                                                                                                                                                                                                                                  ¢                        (  )
                                                                                                                                                                                                                                                            u Substitution : The substitution u= gxwill convert                                                                            fgxgxdx=                                              fudu using 
                                                                                                                                                                                                                                                                                                                                                 ( )                                                 () ()                                          ()
                                                                                                                                                                                                       ¢                                                                                                                                                                                       ( )
                  on  ab,            . Divide  ab,                   into n subintervals of                                      is a function, Fx, such that Fx= fx.  
                         [         ]                    [        ]                                                                                                 ( )                                   ( )              ( )                                                                                                                                                        òò
                                                                                                                                                                                                                                                                                                                                                                                       aga
                                                                                                                                                                                                                                                                                                                                                                                                                                          ()
                  width Dx and choose x* from each interval.                                                                     Indefinite Integral :                             fxdx=+Fxc                                                                               ¢
                                                                                                                                                                                      ( )                    ( )                                             du=gxdx.  For indefinite integrals drop the limits of integration. 
                                                                      i                                                                                                        ò                                                                                             ( )
                                  b                                  ¥                                                                                                                                                                                                    2                                                                                                   28
                                                                                   *                                             where Fx is an anti-derivative of  fx.                                                                                                           23                                                                                                  235
                  Then                                                                         .                                                     ( )                                                             (    )                                 Ex.             5xcos xdx                                                                                            5xcosxdx=                                    cos udu
                                      fxdx=Dlim                            fxx                                                                                                                                                                                                                                                                                                                                                        ()
                                         ()                        å ()                                                                                                                                                                                                                     (      )                                                                                             (      )
                                                                                   i                                                                                                                                                                                                                                                                                                                                       3
                               ò                                                                                                                                                                                                                                      ò                                                                                                    òò
                                  a                        n®¥ i=1                                                                                                                                                                                                      1                                                                                                    11 
                                                                                                                                                                                                                                                                        322                                                                                                                                             8
                                                                                                                                                                                                                                                                                                                                       1                                                              55
                                                                                                                                                                                                                                                             u=xÞdu=3xdxÞ=xdxdu 
                                                                                                                                                                                                                                                                                                                                       3 =sinu =-sin8sin1
                                                                                                                                                                                                                                                                                                                                                                                                                ()                          () ()
                                                                                                                                                                                                                                                                                                                                                                                                                                   (                                )
                                                                                                                                                                                                                                                                                                                                                                                                      33
                                                                                     Fundamental Theorem of Calculus                                                                                                                                                                        331
                  Part I : If  fx is continuous on  ab,                                                  then                        Variants of Part I :                                                                                                    x=1Þu=1=1::xu=2Þ==28 
                                              ( )                                           [        ]                                                                                                                                                       
                                                                                                                                                ux
                                      x                                                                                                d ()                                                                                                                                                                                                                          bb
                                                                                                                                                                              ¢                                                                                                                                                                                                              b
                                                                                                                                                        ftdt=uxféùux  
                                                                                                                                                           ()                   () ()
                   gx= ftdt is also continuous on  ab,                                                                                       ò                                              ëû                                                              Integration by Parts :  udv=-uvvdu and                                                                      udv=-uvvdu.  Choose u and dv from 
                       ()ò ()                                                                            [        ]                   dx a                                                                                                                                                                      òòòò
                                     a                                                                                                                                                                                                                                                                                                                                                       a
                                                                                                                                                                                                                                                                                                                                                                    aa
                                              dx                                                                                       d b                                      ¢                                                                           integral and compute du by differentiating u and compute v using v= dv . 
                                                                                                                                                       ftdt=-vxféùvx  
                               ¢                                                                                                                          ()                     () ()
                  and gx==ftdtfx.                                                                                                                                                                                                                                                                                                                                                                                ò
                                 ()                           ()                  ()                                                         ò                                                ëû
                                                                                                                                               vx
                                                    ò dx()
                                             dx a                                                                                                                                                                                                                             -x                                                                                                            5
                                                                                                                                                ux                                                                                                          Ex.  xe dx 
                  Part II :  fxis continuous on ab,                                               ,  Fx is                             d          ()                                                                                                                  ò                                                                                                        Ex.             lnxdx 
                                           ( )                                        [         ]         (    )                                                              ¢¢                                                                                                                                                                                                         ò
                                                                                                                                                        ftdt=-uxfu(x)vxfvx() 
                                                                                                                                                           ()                   ()[ ] ()[ ]                                                                                                                                                                                                3
                                                                                                                                             ò
                                                                                                                                               vx
                                                                                                                                      dx ()                                                                                                                                                  --xx
                                                                                                                                                                                                                                                                u=xdv=eeÞdu=dxv=-                                                                                                                                                              1
                  an anti-derivative of fx(i.e. Fx=                                                      fxdx)                                                                                                                                                                                                                                                                  u=lnxdv=dxÞdu==dxvx 
                                                                  ( )                    ( )         ò       ( )                                                                                                                                                                                                                                                                                                                               x
                                                                                                                                                                                                                                                                     -x-x-x--xx
                                                                                                                                                                                                                                                                xedx=-xe+edx=-xcee-+                                                                                               555
                               b                                                                                                                                                                                                                             òò 5
                                                                                                                                                                                                                                                                                                                                                                                      lnxdx=xlnx-dx=-xln xx
                  then             fxdx=-FbFa.                                                                                                                                                                                                                                                                                                                                                                                               ( () )
                                      ()                     () ()                                                                                                                                                                                                                                                                                                              òò
                                                                                                                                                                                                                                                                                                                                                                                                                     3
                            ò                                                                                                                                                                                                                                                                                                                                                      333
                              a
                                                                                                                                                                                                                                                                                                                                                                                                    =5ln5--3ln32
                                                                                                                                                                                                                                                                                                                                                                                                                () ()
                                                                                                                 Properties                                                                                                                                  
                       fx±gxdx=±fxdxgxdx                                                                                                 cfxdx=cfxdx, c is a constant 
                          (    )          ( )                      ( )                      (    )                                            (     )                     (     )                                                                           Products and (some) Quotients of Trig Functions 
                   òòò òò
                      bbbbb nm                                                                                                                                                                                                                                                                                                                                                                nm
                                                                                                                                                                                                                                                            For sinxcos xdx we have the following :                                                                        For tanxsec xdx we have the following :  
                         fx±gxdx=±fxdxgxdx cfxdx=cfxdx, c is a constant                                                                                                                                                                                               ò                                                                                                             ò
                             () ()                                       ()                         ()                                           ()                             ()
                   òòòòò
                     aaaaa 1. n odd. Strip 1 sine out and convert rest to  1. n odd. Strip 1 tangent and 1 secant out and 
                      a                                                                                                                  bb                                                                                                                                                              22 convert the rest to secants using 
                          fxdx=0                                                                                                             fxdx= ftdt                                                                                                             cosines using sinxx=-1cos                                           , then use 
                             ()                                                                                                                 ()                          ()
                   ò                                                                                                                  òò
                     a                                                                                                                  aa                                                                                                                                                                                                                                                 22
                      ba bb the substitution ux=cos.                                                                                                                                                                                                                                                                                                                                tanxx=-sec1, then use the substitution 
                         fxdx=- fxdx                                                                                                                                                                                                                        2.  m odd. Strip 1 cosine out and convert rest                                                                          ux=sec . 
                             ()                             ()                                                                               fxdx£                         fxdx
                   òò                                                                                                                            ()                           ()
                     ab òò
                                                                                                                                         aa                                                                                                                                                                22 2. m even. Strip 2 secants out and convert rest 
                                                                                             ba                                                                                                                                                                     to sines using cosxx=-1sin                                           , then use 
                  If  fx³gx ona££xbthen                                                          fxdx³ gxdx                                                                                                                                                                                                                                                                                                                     22
                            ( )             (    )                                                  ()                          ()                                                                                                                                  the substitution ux= sin                              .                                                        to tangents using secxx=+1tan                                               , then 
                                                                                          òò
                                                                                             ab                                                                                                                                                             3.  n and m both odd. Use either 1. or 2.                                                                              use the substitution ux= tan                                   . 
                  If fx³0 on a££xb then  b fxdx³0                                                                                                                                                                                                           4.  n and m both even. Use double angle                                                                        3.  n odd and m even. Use either 1. or 2.  
                           ( )                                                      òa       ()
                                                                                                                           b                                                                                                                                        and/or half angle formulas to reduce the                                                               4.  n even and m odd. Each integral will be 
                  If m££fxMon a££xb then mb-a£fxdx£-Mba                                                                                                                                                                                                             integral into a form that can be integrated.                                                                   dealt with differently. 
                                     ( )                                                            (           )                ()                      ( )
                                                                                                                        ò
                                                                                                                          a                                                                                                                                                                                                                                       2               1                                        2              1
                                                                                                                                                                                                                                                            Trig Formulas : sin2x=2sinxxcos                                                             ,  cosxx=+1cos2 , sinxx=-1cos2  
                                                                                                                                                                                                                                                                                                       (       )                 ( )            ( )                  ( )          2 (                (       ))               ( )         2 (                (       ))
                                                                                                        Common Integrals                                                                                                                                     
                                                                                                                                                                                                                                                                                35                                                                                                           sin5 x
                   òkdx=+kxc                                                                      òcosudu=+sinuc                                                      òtanudu=+lnsecuc                                                                      Ex. òtanxsec xdx                                                                                                   Ex. ò              3    dx 
                                                                                                                                                                                                                                                                                                                                                                                             cos x
                         nn+1
                                         1                                                                                                                                                                                                                             3524                                                                                                             5422
                                                                                                                                                                                                                                                                                                                                                                                                                                                         sin x
                   òxdx=x+cn,1¹-                                                                  òsinudu=-+cosuc                                                     òsecudu=lnsecu++tanuc                                                                     tanxsecxdx=tanxsecxtanxsecxdx                                                                                       sinxsinxxsin                                           (sin)x
                                       n+1                                                                                                                                                                                                                   òò dx==dxdx
                                                                                                                                                                                                                                                                                                                                                                                         333
                                                                                                                                                                                                                                                                                                                                                                                òòò
                         -1                1                                                                2                                                                 1                  1         -1     u                                                                                                                                                                 cosxcosxxcos
                      xdx=dx=+ln xc                                                                  secudu=+tanuc                                                                    duc=+tan                                   24                                                                                                                                                                                    22
                                                                                                                                                                                                               (     )                                                               =-secx1secxtanxsecxdx                                                                                                                    sin x
                                           x                                                                                                                                22 aa                                                                                                                                                                                                                           (1-cos)x
                   òò ò                                                                                                                                               ò                                                                                                                     (                    )
                                                                                                                                                                         au+                                                                                                             ò ==dxuxcos
                                                                                                                                                                                                                                                                                                                                                              3 ( )
                                                                                                                                                                                                                                                                                                                                                                                                         ò
                          11                                                                         secutanudu=+secuc                                                         1                         -1     u                                                                                24                                                                                                                 cos x                                                      
                                 dx=lnax++bc                                                                                                                                            duc=+sin
                   ò                        a                                                     ò                                                                                                          (     )                                                                 =u-=1uduuxsec                                                                                                                    22
                      axb+                                                                                                                                            ò       22                                a                                                                           (             )                          (                   )                                                     (1)-u24
                                                                                                                                                                            au-                                                                                                          ò                                                                                                                                                    12-+uu
                                                                                                                                                                                                                                                                                                                                                                                                    =-du=-                                                      du
                                                                                                                                                                                                                                                                                                                                                                                                                      33
                                                                                                                                                                                                                                                                                                                                                                                                            òò
                                                                                                     cscucotudu=-+cscuc                                                                                                                                                                            75                                                                                                              uu
                      lnudu=uln u-+uc                                                                                                                                                                                                                                                     11
                   ò                             ( )                                              ò                                                                                                                                                                                  =secx-+sec xc
                                                                                                                                                                                                                                                                                          75                                                                                                                       22
                                                                                                                                                                                                                                                                                                                                                                                                         11
                                                                                                            2                                                                                                                                                                                                                                                                                       =secx+2lncosx-+cos xc
                        uu                                                                           cscudu=-+cotuc                                                                                                                                                                                                                                                                                      22
                   òeeduc=+                                                                       ò
                  Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.                                                                                                       © 2005 Paul Dawkins                                      Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.                                                                                                       © 2005 Paul Dawkins 
                                                          Calculus Cheat Sheet                                                                                                                       Calculus Cheat Sheet 
           Trig Substitutions : If the integral contains the following root use the given substitution and                                                                                           Applications of Integrals 
           formula to convert into an integral involving trig functions.                                                                                               b
                                                                                                                                                      Net Area :          fxdx represents the net area between  fx and the 
                                                                                                                                                                           ()                                                   ()
                                                                                                                                                                      ò
                  222                    a                   222 a                                       222                   a                                       a
                a-bxxÞ=sinq  bx-axÞ=secq  a+bxxÞ=tanq 
                                         b                                          b                                          b                      x-axis with area above x-axis positive and area below x-axis negative. 
                        22 22 22
                    cosqq=-1sin                                 tanqq=-sec1                                secqq=+1tan             
                                                                                                                                                                                                                                                                                       
                       16                                                           16        2                   12
           Ex.               dx                                             ó cosqddqq=
                 ò2                                                                         (         )
                           2                                                                  3                    2
                   x49-x                                                        4  2                          ò                                       Area Between Curves : The general formulas for the two main cases for each are, 
                                                                            õ sinq
                                                                                 sinqq2cos
                                                                                9     (    )                                                                                    b                                                                  d
                22
           x=sinqÞ=dxdcosqq                                                                                 2                                                                    éù                                                                  éù
                                                                                                                                                                                                   éù                                                                éù
                                                                                                                                                       y=fxÞA=-upper  functionlower  function dx &  x=fyÞA=-right  functionleft  function dy 
                33 =12cscdcqq=-+12cot                                                                                                                         ()                 ëûëû() ëûëû
                                                                                                                                                                              ò                                                                  ò
                                                                                                   ò                                                                           a                                                                  c
                     222
                       =4-4sinq==4cosqq2cos  
              49-x                                                          Use Right Triangle Trig to go back to x’s. From                           If the curves intersect then the area of each portion must be found individually.  Here are some 
                        2                                                   substitution we have sinq = 3x  so,                                       sketches of a couple possible situations and formulas for a couple of possible cases. 
           Recall      xx=.  Because we have an indefinite                                                       2
           integral we’ll assume positive and drop absolute 
           value bars.  If we had a definite integral we’d 
           need to compute q’s and remove absolute value 
           bars based on that and,                                                                                       
                                       xxif 0³                                                                          2
                                    ì                                       From this we see that cotq = 49- x .  So, 
                               x=  
                                    í                                                                                3x
                                      -0 Ex. Axis :  ya=£0 Ex. Axis : ya=>0 Ex. Axis : ya=£0 
                                                             2                                                                 k
                      ax++bxc                                    ax++bxc 
                                               2                ( )                                                2
                                            ax++bxc                                    ax++bxc                 ax++bxc
                                                                                                              (               )
            
                      2                                                         2                              2
                    7xx+13                                                    7xx+13        A BxC+ Ax(++4)(Bx+-Cx)()1
           Ex.                dx                                                        =+=  
                         2                                                         222
                 ò                                                                         x-1
                   (xx-+14)()                                                (x-1)(x+4)x+4(xx-+14)()
                 2
               7xx+13             4     3x+16                               Set numerators equal and collect like terms. 
                    2dx=+dx
                                         2
                                 x-1
           òò
              (xx-+14)()                x+4                                        22
                                                                                7x+134x=A+Bx+C-Bx+-AC 
                                                                                               ()()
                             4     3x 16
                       =++dx                                           
                            x-1    22
                          ò Set coefficients equal to get a system and solve 
                                  xx++44
                                                21-
                                        3 xto get constants.                                                                                                                                                                                                                           
                       =4lnxx-1+ln++48tan ()
                                             ( )
                                        22                                                                                                                                                                             radius :ay- 
                                                                                 A+B=7C-B=1340AC-=                                                    outer radius :a-fx outer radius: a+gx                                                             radius : ay+ 
           Here is partial fraction form and recombined.                                                                                () ()
                                                                                                                                                                                                                       width :  fy-gy 
                                                                                    A=4BC==316                                                                                                                                      ()()
                                                                                                                                                      inner radius :  a-gx inner radius: a+fx                                                           width :  fy-gy 
                                                                                                                                                                               ()                              ()                                                    ()()
                                                                                                                                                       
           An alternate method that sometimes works to find constants. Start with setting numerators equal in                                         These are only a few cases for horizontal axis of rotation.  If axis of rotation is the x-axis use the 
                                        22
           previous example : 7x+13x=Ax+41+Bx+-Cx .  Chose nice values of x and plug in. 
                                                                  ()()
                                                      ()                                                                                               ya=£0 case with a=0. For vertical axis of rotation (xa=>0 and xa=£0) interchange x and 
           For example if x=1 we get 205=A which gives A=4.  This won’t always work easily.                                                           y to get appropriate formulas. 
           Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.                               © 2005 Paul Dawkins               Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.                                © 2005 Paul Dawkins 
                                                          Calculus Cheat Sheet 
           Work : If a force ofFxmoves an object                               Average Function Value : The average value 
                                        ( )
                                                         b                                                               1b
                                                                               of  fx on a££xb is f=                            fxdx 
                                                                                      ( )                       avg         ò     ()
           ina££xb, the work done is W=                    Fxdx                                                        ba-   a
                                                       òa()
            
           Arc Length  Surface Area : Note that this is often a Calc II topic. The three basic formulas are, 
                  b                      b                                                      b
           L=       ds          SA=       2pyds (rotate about x-axis)                  SA=       2pxds (rotate about y-axis) 
                 òa                    òa                                                     òa
           where ds is dependent upon the form of the function being worked with as follows. 
                         dy 2                                                   dx 2 dy 2
           ds=1+dx  if  ,y=fxa££xb  ds=+dt if  x=ft,,y=gta££tb
                       (    )                  ()                             () ( )                           ()          ()
                         dx                                                     dt        dt
                         dx 2                                                   2dr 2
                                                                      ds=r+dq if  ,r=fqqab££
           ds=1+dy  if  ,x=fya££yb                                                  ( )                     ()
                       ( )                     ()                                     dq
                         dy
           With surface area you may have to substitute in for the x or y depending on your choice of ds to 
           match the differential in the ds.  With parametric and polar you will always need to substitute. 
            
                                                              Improper Integral 
           An improper integral is an integral with one or more infinite limits and/or discontinuous integrands. 
           Integral is called convergent if the limit exists and has a finite value and divergent if the limit 
           doesn’t exist or has infinite value. This is typically a Calc II topic. 
            
           Infinite Limit 
                  ¥                     t                                              bb
           1.        fxdx=lim fxdx  2.                                                    fxdx=lim              fxdx
                       ()                   ()                                              ()                    ()
                òò òò
                  aa -¥ t
                                  t®¥                                                                 t®-¥
                  ¥¥c
           3.        fxdx=+fxdxfxdx provided BOTH integrals are convergent. 
                       ()                ()               ()
                òòò
                  --¥¥c
           Discontinuous Integrand 
                                   bb bt
           1.  Discont. at a:        fxdx=lim              fxdx 2.  Discont. at b :                        fxdx=lim             fxdx
                                       ()                    ()                                              ()                   ()
                                 òò òò
                                                      +                                                                    -
                                   at aa
                                                  ta®                                                                  tb®
                                                     bcb
           3.  Discontinuity at a< 0 then         ¥ 1 dx converges if  p >1 and diverges for  p £1. 
                                             òa xp
            
                                                    Approximating Definite Integrals 
                                     b                                                                                   ba-
           For given integral          fxdx and a n (must be even for Simpson’s Rule) define D=x                               and 
                                   òa()                                                                                   n
           divide  ab,      into n subintervals  xx, ,  xx, , … ,  xx,                       with  xa=       and  xb=       then, 
                    [    ]                          [       ]  [       ]        [         ]
                                                       01 12                       n-1   n           0              n
                                    b                                                           *
                                                             ***
                                                      éù
           Midpoint Rule :            fxdx»Dxfx+fx++Lfx , x is midpoint  xx,                                                 
                                        ()                (   )      (   )           (   )                       [        i ]
                                                             12ni                                                   i-1
                                  ò
                                   a ëû
                                     b              Dx
           Trapezoid Rule :            fxdx»éùfx+2fx++22fx+L++fxfx  
                                         ()                 () () ()                                    ( ) ()
                                                               0121nn-
                                   òa 2ëû
                                     b              Dx
           Simpson’s Rule :            fxdx»éùfx+4fx+2fx+L+24fx++fxfx  
                                         ()                 () () ()                                  ( ) ( ) ()
                                                               012n--21nn
                                                        ëû
                                   ò
                                     a3
           Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes.                               © 2005 Paul Dawkins 
The words contained in this file might help you see if this file matches what you are looking for:

...Calculus cheat sheet integrals standard integration techniques definitions note that at many schools all but the substitution rule tend to be taught in a ii class definite integral suppose fx is continuous anti derivative an of bgb u gxwill convert fgxgxdx fudu using on ab divide into n subintervals function such oo aga width dx and choose x from each interval indefinite fxdx fxc du gxdx for drop limits i o b where then ex xcos xdx xcosxdx cos udu dlim fxx xdxdu...

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