Find the integral: ∫ cos 3 ⁡ x d x

Question

Find the integral:  ∫      cos          3        ⁡  x  d  x

Jackson Trevino · Accepted Answer

Use Pythagorean Identities:       cos          2        ⁡  x  =  1  −      sin          2        ⁡  x  ∫  (  1  −      sin          2        ⁡  x  )  cos  ⁡  x  d  xUse Integration by Substitution.Let   u  =  sin  ⁡  x  ,  d  u  =  cos  ⁡  x  d  xUsing   u and   du above, rewrite   ∫  (  1  −      sin          2        ⁡  x  )  cos  ⁡  x  x  d  ∫  1  −      u          2        d  uUse Power Rule:   ∫      x          n        d  x  =            x              n        +        1                    n      +      1        +  C  u  −            u              3              3  Substitute   u  =  sin  ⁡  x back into the original integral.  sin  ⁡  x  −                    sin                  3                    ⁡      x        3  Add constant.  sin  ⁡  x  −                    sin                  3                    ⁡      x        3    +  C

Find the integral: int cos^3 for xdx

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