Calculate: ∫ 0 π cos ⁡ ( x ) log ⁡ ( sin 2 ⁡ ( x ) + 1 ) d x

Question

Calculate:      ∫          0              π        cos  ⁡  (  x  )  log  ⁡  (      sin    2    ⁡  (  x  )  +  1  )  d  x

sepolturamo · Accepted Answer

Behold the power of symmetry:  cos  ⁡  (  π  −  x  )  =  −  cos  ⁡  x    and    sin  ⁡  (  π  −  x  )  =  sin  ⁡  x  ,therefore                              ∫          0          π                cos        ⁡        x        log        ⁡        (                  sin          2                ⁡        x        +        1        )                d        x                            =                  ∫          0          π                cos        ⁡        (        π        −        u        )        log        ⁡        (                  sin          2                ⁡        (        π        −        u        )        +        1        )                d        u                                          =        −                  ∫          0          π                cos        ⁡        u        log        ⁡        (                  sin          2                ⁡        u        +        1        )                d        u        ,            hence the integral evaluates to 0.

raulgallerjv · Accepted Answer

∫  cos  ⁡  x  ln  ⁡  (  1  +      sin    2    ⁡  x  )     d  x  =  ln  ⁡  (  1  +      sin    2    ⁡  x  )  ∫  cos  ⁡  x     d  x  −  ∫      (                            d                     ln          ⁡          (          1          +                      sin            2                    ⁡          x          )                          d          x                      ∫    cos    ⁡    x         d    x    )    d  x  =  sin  ⁡  x  ⋅  ln  ⁡  (  1  +      sin    2    ⁡  x  )  −  ∫                    2                  sin          2                ⁡        x        cos        ⁡        x                    1        +                  sin          2                ⁡        x              d  x  ∫                    2                  sin          2                ⁡        x        cos        ⁡        x                    1        +                  sin          2                ⁡        x              d  x  =  ∫                    2        (        1        +                  sin          2                ⁡        x        −        1        )        cos        ⁡        x                    1        +                  sin          2                ⁡        x              d  x  =  2  ∫  cos  ⁡  x     d  x  −  2  ∫                    cos        ⁡        x                    1        +                  sin          2                ⁡        x              d  xSet sinx=u for the last integral

Calculate: <msubsup> ∫ <mrow class="MJX-TeXAtom-ORD"> 0

Answered question

Answer & Explanation

New Questions in Calculus 2