Integrating ∫ − π π cos 2 ⁡ ( x ) | cos ⁡ ( x ) | | sin ⁡ ( x ) | d x ∫ − π π cos 2 ⁡ ( x ) | cos ⁡ ( x ) | d x

Question

Integrating                     ∫                  −          π                          π                            cos        2            ⁡      (      x      )              |            cos      ⁡      (      x      )              |                    |            sin      ⁡      (      x      )              |            d      x                      ∫                  −          π                          π                            cos        2            ⁡      (      x      )              |            cos      ⁡      (      x      )              |            d      x

Krish Finley · Accepted Answer

For both integrals you can use substitution.For the top:Let   u  =  cos  ⁡  x, so   d  u  =  −  sin  ⁡  x  d  xWhen x=0 then u=1.When   x  =      π    2   then u=0.      ∫    0                  π        2              c  o      s    3    x  sin  ⁡  x  d  x  =  −      ∫    1    0        u    3    d  u  =  −      1    4        u    4                      |              1    0    =      1    4  For the bottom:Let   u  =  sin  ⁡  x so   d  u  =  cos  ⁡  x  d  xWhen x=0 then u=0.When   x  =      π    2   then u=1.      ∫    0          π      2            cos    3    ⁡  x  d  x  =      ∫    0          π      2        (  1  −      sin    2    ⁡  x  )  cos  ⁡  x  d  x  =      ∫    0    1    1  −      u    2    d  u  =  u  −            u      3        3                      |              0    1    =      2    3  So the fraction is:            1      4              2      3        =      3    8

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