Antiderivative of a trigonometric integral ∫ sin 3 ⁡ ( x ) ( cos 4 ⁡ ( x ) + 3 cos 2 ⁡ ( x ) + 1 ) ⋅ arctan ⁡ ( sec ⁡ ( x ) + cos ⁡ ( x ) ) I am unable to manipulate this integral. The actual integral in question is definite with limits from 0 to π 2 , but I feel it can't be calculated without knowing the antiderivative.My initial try was to take sec ⁡ ( x ) + cos ⁡ ( x ) as t, but It didn't solve.

Question

Antiderivative of a trigonometric integral  ∫                    sin        3            ⁡      (      x      )              (              cos        4            ⁡      (      x      )      +      3              cos        2            ⁡      (      x      )      +      1      )      ⋅      arctan      ⁡      (      sec      ⁡      (      x      )      +      cos      ⁡      (      x      )      )      I am unable to manipulate this integral. The actual integral in question is definite with limits from 0 to       π    2  , but I feel it can&#039;t be calculated without knowing the antiderivative.My initial try was to take   sec  ⁡  (  x  )  +  cos  ⁡  (  x  ) as t, but It didn&#039;t solve.

Haylie Cherry · Accepted Answer

Instead try                          t        =                  tan                      −            1                          ⁡        (        sec        ⁡        x        +        cos        ⁡        x        )                                  d        t        =                              sec            ⁡            x            tan            ⁡            x            −            sin            ⁡            x                                (            sec            ⁡            x            +            cos            ⁡            x                          )              2                        +            1                          d        x                                          =                              sin            ⁡            x            (                          sec              2                        ⁡            x            −            1            )                                              cos              2                        ⁡            x            +            3            +                          sec              2                        ⁡            x                          d        x        =                                            sin              3                        ⁡            x                                              cos              4                        ⁡            x            +            3                          cos              2                        ⁡            x            +            1                          d        x            Then the integral simply becomes  ∫      1    t    d  t  =  log  ⁡  t  =  log  ⁡  arctan  ⁡  (  sec  ⁡  x  +  cos  ⁡  x  )

Antiderivative of a trigonometric integral ∫ sin

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