Evaluating the trigonometric integral ∫ cos ⁡ x 1 + cos ⁡ x d x

Question

Evaluating the trigonometric integral   ∫            cos      ⁡      x              1      +      cos      ⁡      x          d  x

gutinyalk · Accepted Answer

Hint Here&#039;s one method: Rewrite the integrand as            cos      ⁡      x              1      +      cos      ⁡      x        =  1  −      1          1      +      cos      ⁡      x        ,so that the integral becomes  ∫            cos      ⁡      x              1      +      cos      ⁡      x          d  x  =  ∫      (    1    −          1              1        +        cos        ⁡        x              )    d  x  =  x  −      ∫                  d        x                    1        +        cos        ⁡        x              .Now, the remaining integral can be handled by exploiting the Pythagorean identity:      1          1      +      cos      ⁡      x        =      1          1      +      cos      ⁡      x        ⋅            1      −      cos      ⁡      x              1      −      cos      ⁡      x        =            1      −      cos      ⁡      x              1      −              cos        2            ⁡      x        =            1      −      cos      ⁡      x                      sin        2            ⁡      x        .Additional hint Now,      ∫                  1        −        cos        ⁡        x                              sin          2                ⁡        x              d    x    =  ∫            d      x                      sin        2            ⁡      x        −  ∫            cos      ⁡      x                      sin        2            ⁡      x          d  x  .The first integral is elementary, and the second can be handled with a straightforward substitution.

Dayanara Terry · Accepted Answer

Hint   cos  ⁡  (  x  )  =            1      −              tan        2            ⁡      (      x              /            2      )              1      +              tan        2            ⁡      (      x              /            2      )       so the integral reduces to   0.5  ∫  (  1  −      tan    2    ⁡  (  x      /    2  )  )  d  x where we can then use identity       tan    2    ⁡  (  x      /    2  )  =      sec    2    ⁡  (  x      /    2  )  −  1 to get the final integral which is   x  −  tan  ⁡  (  x      /    2  )  +  C

Evaluating the trigonometric integral ∫ cos ⁡<!-

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