Find the integral: ∫ cot 3 ⁡ x d x

Question

Find the integral:  ∫      cot          3        ⁡  x  d  x

Pignatpmv · Accepted Answer

Use Pythagorean Identities:       cot          2        ⁡  x  =      csc          2        ⁡  x  −  1  ∫  (      csc          2        ⁡  x  −  1  )  cot  ⁡  x  d  xExpand.  ∫  cot  ⁡  x      csc          2        ⁡  x  −  cot  ⁡  x  d  xUse Sum Rule:  ∫  f  (  x  )  +  g  (  x  )  d  x  =  ∫  f  (  x  )  d  x  +  ∫  g  (  x  )  d  x  .  ∫  cot  ⁡  x      csc          2        ⁡  x  d  x  −  ∫  cot  ⁡  x  d  xSimplify the trigonometric functions.  ∫            cos      ⁡      x                      sin                  3                    ⁡      x        d  x  −  ∫  cot  ⁡  x  d  xUse Integration by Substitution on   ∫            cos      ⁡      x                      sin                  3                    ⁡      x        d  xLet   u  =  sin  ⁡  x  ,  d  u  =  cos  ⁡  x  d  xUsing   u and   du above, rewrite   ∫            cos      ⁡      x                      sin                  3                    ⁡      x        d  x.  ∫      1          u              3              d  uUse Power Rule:   ∫      x          n        d  x  =            x              n        +        1                    n      +      1        +  C.  −      1          2              u                  2                    Substitute   u  =  sinx back into the original integral.  −      1          2              sin                  2                    ⁡      x       Rewrite the integral with the completed substitution.  −      1          2              sin                  2                    ⁡      x        −  ∫  cot  ⁡  x  d  xUse Trigonometric Integration: the integral of   cot  ⁡  x of   ln  ⁡  (  sin  ⁡  x  )  −      1          2              sin                  2                    ⁡      x        −  ln  ⁡  (  sin  ⁡  x  )Add constant.  −      1          2              sin                  2                    ⁡      x        −  ln  ⁡  (  sin  ⁡  x  )  +  C

Find the integral: int cot^(3) xdx

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