Compute ∫ 0 π / 2 cos ⁡ x 2 − sin ⁡ 2 x d x

Question

Compute       ∫    0          π              /            2                  cos      ⁡              x                    2      −      sin      ⁡              2        x              d  x

Kathryn Moody · Accepted Answer

Hint:Knowing that   sin  ⁡  2  x  =  2  sin  ⁡  x  cos  ⁡  x and       sin    2    ⁡  x  +      cos    2    ⁡  x  =  1. The integral can be expressed as  I  =      ∫    0          π              /            2                  cos      ⁡      x              1      +      (      sin      ⁡      x      −      cos      ⁡      x              )        2                 d  xthen use substitution   x  ↦      π    2    −  x, we have  I  =      ∫    0          π              /            2                  sin      ⁡      x              1      +      (      sin      ⁡      x      −      cos      ⁡      x              )        2                 d  xAdd the two I&#039;s and let   u  =  sin  ⁡  x  −  cos  ⁡  x

Cierra Castillo · Accepted Answer

I                            =                  ∫          0                      π                          /                        2                                                cos            ⁡                          x                                            2            −            sin            ⁡                          2              x                                      d        x                                          =                  ∫          0                      π                          /                        2                                                cos            ⁡                          x                                            2            −            2            sin            ⁡            x            cos            ⁡            x                          d        x                                          =                  ∫          0                      π                          /                        2                                                cos            ⁡                          x                                            1            +                          cos              2                        ⁡            x            −            2            sin            ⁡            x            cos            ⁡            x            +                          sin              2                        ⁡            x                          d        x                                          =                  ∫          0                      π                          /                        2                                                cos            ⁡                          x                                            1            +            (            cos            ⁡            x            −            sin            ⁡            x                          )              2                                      d        x                                          =                  1          2                          (                      ∫            0                          π                              /                            2                                                          cos              ⁡                              x                                                    1              +              (              cos              ⁡              x              −              sin              ⁡              x                              )                2                                              d          x          +                      ∫            0                          π                              /                            2                                                          sin              ⁡                              x                                                    1              +              (              cos              ⁡              x              −              sin              ⁡              x                              )                2                                              d          x          )                                                  =                  1          2                          ∫          0                      π                          /                        2                                                cos            ⁡                          x                        +            sin            ⁡                          x                                            1            +            (            cos            ⁡            x            −            sin            ⁡            x                          )              2                                      d        x                                          =        −                  1          2                          ∫          0                      π                          /                        2                                                d            (            cos            ⁡                          x                        −            sin            ⁡                          x                        )                                1            +            (            cos            ⁡            x            −            sin            ⁡            x                          )              2                                                                        =        −                  1          2                arctan        ⁡        (        cos        ⁡                  x                −        sin        ⁡                  x                )                              |                                0                                              π              2                                                                        =                  π          4

Compute <msubsup> ∫ 0 <mrow class="MJX-TeXAtom-ORD"> &#x0

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