How do we handle the integral I 4 = ∫ ( cos ⁡ θ 1 + sin 2 ⁡ θ ) 4 d θ?

Question

How do we handle the integral       I          4        =  ∫            (                        cos          ⁡          θ                          1          +                      sin                          2                                ⁡          θ                    )              4        d  θ?

teleriasacr · Accepted Answer

In general, for any even power, substitute       tan    2    ⁡  θ  =      1    2        tan    2    ⁡  x to transform the integral to a manageable one      I          2      m        =  ∫            (                        cos          ⁡          θ                          1          +                      sin                          2                                ⁡          θ                    )              2      m        d  θ  =      1          2              m        −                  1          2                      ∫      cos          2      m        ⁡  x    (  1  +      cos    2    ⁡  x      )          m      −      1        d  xwhich, for   m  =  2, reduces to                              I          4                =                                      1                      2                          3                              /                            2                                      ∫                  cos                      4                          ⁡        x                (        1        +                  cos          2                ⁡        x        )                d        x                            =                                      1                      2                          5                              /                            2                                                (                      1            3                                cos            5                    ⁡          x          +                      11            12                                cos            3                    ⁡          x          +                      11            8                    cos          ⁡          x          )                sin        ⁡        x        +                  11                      2                          11                              /                            2                                      x        +        C

How do we handle the integral I_4=int ((cos theta)/(1+sin^2 theta))^4 d theta?

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