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File: Study Pdf 89087 | C2 Trigint
1 6 trigonometric integrals and trig substitutions 26 1 6 trigonometric integrals and trigonometric substitutions 1 6 1 trigonometricintegrals herewediscussintegralsofpow ers of trigonometric functions to that end the following half ...

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                                           1.6. TRIGONOMETRIC INTEGRALS AND TRIG. SUBSTITUTIONS                                26
                                             1.6. Trigonometric Integrals and Trigonometric
                                                                           Substitutions
                                       1.6.1. TrigonometricIntegrals. Herewediscussintegralsofpow-
                                  ers of trigonometric functions. To that end the following half-angle
                                  identities will be useful:
                                                                   sin2 x = 1(1 −cos2x),
                                                                               2
                                                                  cos2x = 1(1+cos2x).
                                                                               2
                                       Remember also the identities:
                                                                         2           2
                                                                     sin x+cos x = 1,
                                                                         2                 2
                                                                     sec x = 1+tan x.
                                       1.6.1.1. Integrals of Products of Sines and Cosines. We will study
                                  now integrals of the form
                                                                     Z       m        n
                                                                         sin   xcos xdx,
                                  including cases in which m = 0 or n = 0, i.e.:
                                                             Z cosnxdx;              Z sinmxdx.
                                       The simplest case is when either n = 1 or m = 1, in which case the
                                  substitution u = sinx or u = cosx respectively will work.
                                       Example: Z sin4xcosxdx = ···
                                  (u = sinx, du = cosxdx)
                                                               Z                5                5
                                                       · · · =    u4du = u +C = sin x +C .
                                                                               5                5
                                       More generally if at least one exponent is odd then we can use the
                                                 2          2
                                  identity sin x+cos x = 1totransformtheintegrandintoanexpression
                                  containing only one sine or one cosine.
                            1.6. TRIGONOMETRIC INTEGRALS AND TRIG. SUBSTITUTIONS     27
                          Example:
                              Z    2 xcos3xdx = Z sin2xcos2xcosxdx
                                 sin
                                              =Z sin2x(1−sin2x)cosxdx=···
                       (u = sinx, du = cosxdx)
                                  · · · = Z u2 (1 − u2)du = Z (u2 − u4)du
                                                        u3   u5
                                                      = 3 − 5 +C
                                                            3    sin5
                                                      = sin x −      x +C .
                                                           3       5
                          If all the exponents are even then we use the half-angle identities.
                          Example:
                               Z    2     2       Z 1           1
                                  sin xcos xdx =    2(1 −cos2x)2(1+cos2x)dx
                                               =1Z           2 2x)dx
                                                  4 Z (1 −cos
                                               =1 (1−1(1+cos4x))dx
                                                  4       2
                                               =1Z (1−cos4x)dx
                                                  8
                                               =x−sin4x+C.
                                                  8    32
                          1.6.1.2. Integrals of Secants and Tangents. The integral of tanx
                       can be computed in the following way:
                       Z tanxdx=Z sinx dx=−Z du = −ln|u|+C =−ln|cosx|+C ,
                                      cosx           u
                       where u = cosx. Analogously
                          Z cotxdx = Z cosx dx = Z du = ln|u|+C = ln|sinx|+C ,
                                        sinx         u
                       where u = sinx.
                            1.6. TRIGONOMETRIC INTEGRALS AND TRIG. SUBSTITUTIONS     28
                          The integral of secx is a little tricky:
                         Z secxdx = Z secx(tanx+secx)dx = Z secxtanx+sec2xdx=
                                           secx+tanx                secx+tanx
                                            Z du =ln|u|+C = ln|secx+tanx|+C ,
                                               u
                                                                 2
                       where u = secx+tanx, du = (secxtanx+sec x)dx.
                          Analogously:
                                     Z cscxdx = −ln|cscx+cotx|+C .
                          More generally an integral of the form
                                              Z     m     n
                                                 tan xsec xdx
                       can be computed in the following way:
                           (1) If m is odd, use u = secx, du = secxtanxdx.
                                                               2
                           (2) If n is even, use u = tanx, du = sec xdx.
                                   Z     3    2
                          Example:    tan xsec xdx = ···
                          Since in this case m is odd and n is even it does not matter which
                       method we use, so let’s use the first one:
                          (u = secx, du = secxtanxdx)
                                Z     2                     Z   2
                           · · · = tan x secx tanxsecxdx =     (u −1)udu
                                   | {z }|{z}|     {z    }
                                    u2−1   u       du       Z
                                                          = (u3−u)du
                                                              4   u2
                                                          =u − +C
                                                             4    2
                                                          = 1sec4x− 1sec2x+C .
                                                             4        2
                          Next let’s solve the same problem using the second method:
                            1.6. TRIGONOMETRIC INTEGRALS AND TRIG. SUBSTITUTIONS     29
                          (u = tanx, du = sec2xdx)
                             Z     3    2       Z   3     u4 +C = 1     4
                                tan x sec xdx =    u du = 4         4 tan x+C .
                               | {z }|  {z  }
                                 u3      du
                       Although this answer looks different from the one obtained using the
                       first method it is in fact equivalent to it because they differ in a con-
                       stant:
                                 1   4    1    2     2   1   4    1   2    1
                                 4 tan x = 4(sec x−1) = 4 sec x− 2 sec x+4 .
                                                         |      {z      }
                                                            previous answer
                          1.6.2. Trigonometric Substitutions. Here we study substitu-
                       tions of the form x = some trigonometric function.
                          Example: Find Z √1−x2dx.
                          Answer: We make x = sint, dx = costdt, hence
                                    √     2   p       2    √ 2
                                      1−x = 1−sin t= cos t=cost,
                       and             Z
                        Z √1−x2dx=Z cost costdt
                                            2
                                     = cos tdt
                                     =Z 1(1+cos2t)dt                (half-angle identity)
                                         2
                                     = t + sin2t +C
                                       2     4
                                     = t + 2sintcost +C          (double-angle identity)
                                       2       4
                                               p       2
                                     = t + sint  1−sin t +C
                                       2          2
                                          −1      √      2
                                     = sin  x + x 1−x +C .
                                          2          2
                          The following substitutions are useful in integrals containing the
                       following expressions:
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...Trigonometric integrals and trig substitutions trigonometricintegrals herewediscussintegralsofpow ers of functions to that end the following half angle identities will be useful sin x cosx remember also cos sec tan products sines cosines we study now form z m n xcos xdx including cases in which or i e cosnxdx sinmxdx simplest case is when either substitution u sinx respectively work example sinxcosxdx du cosxdx udu c more generally if at least one exponent odd then can use identity totransformtheintegrandintoanexpression containing only sine cosine xcosxdx sinxcosxcosxdx all exponents are even dx secants tangents integral tanx computed way tanxdx ln where analogously cotxdx secx a little tricky secxdx secxtanx cscxdx cscx cotx an xsec secxtanxdx since this it does not matter method so let s rst tanxsecxdx next solve same problem using second although answer looks dierent from obtained fact equivalent because they dier con stant previous here substitu tions some function find make sint ...

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