Integrating trig function ∫ ( sin 2 ⁡ x ) 2 + ( 2 sin ⁡ x cos ⁡ x ) 2 d x = ∫ sin 2 ⁡ x sin 2 ⁡ x + 4 sin 2 ⁡ x cos 2 ⁡ x d x

Question

Integrating trig function   ∫            (              sin        2            ⁡      x              )        2            +      (      2      sin      ⁡      x      cos      ⁡      x              )        2              d    x    =  ∫                    sin        2            ⁡      x              sin        2            ⁡      x      +      4              sin        2            ⁡      x              cos        2            ⁡      x        d    x

Stacy Johns · Accepted Answer

∫                                            sin              2                        ⁡            x                          sin              2                        ⁡            x            +            4                          sin              2                        ⁡            x                          cos              2                        ⁡            x                    d          x                                                  =        ∫                                            sin              2                        ⁡            x            (                          sin              2                        ⁡            x            +            4                          cos              2                        ⁡            x            )                    d          x                                                  =                  {                                                    ∫                                  sin                  ⁡                  x                                      (                                          sin                      2                                        ⁡                    x                    +                    4                                          cos                      2                                        ⁡                    x                    )                                    d                  x                                                                                                      ,                sin                ⁡                x                ≥                0                                                                    −                ∫                                  sin                  ⁡                  x                                      (                                          sin                      2                                        ⁡                    x                    +                    4                                          cos                      2                                        ⁡                    x                    )                                    d                  x                                                                                                      ,                sin                ⁡                x                &amp;lt;                0                                                                                                =                  {                                                    ∫                                  sin                  ⁡                  x                                      (                    1                    +                    3                                          cos                      2                                        ⁡                    x                    )                                    d                  x                                                                                                                                  ,                sin                ⁡                x                ≥                0                                                                    −                ∫                                  sin                  ⁡                  x                                      (                    1                    +                    3                                          cos                      2                                        ⁡                    x                    )                                    d                  x                                                                                                                                  ,                sin                ⁡                x                &amp;lt;                0                                                                                                =                  {                                                    −                                  1                                      3                                                  ∫                                  sin                  ⁡                  x                                      (                    1                    +                    3                                          cos                      2                                        ⁡                    x                    )                                    d                  (                                      3                                    cos                  ⁡                  x                  )                                                            ,                sin                ⁡                x                ≥                0                                                                                      1                                      3                                                  ∫                                  sin                  ⁡                  x                                      (                    1                    +                    3                                          cos                      2                                        ⁡                    x                    )                                    d                  (                                      3                                    cos                  ⁡                  x                  )                                                            ,                sin                ⁡                x                &amp;lt;                0                                                                  Now say   z  =      3    cos  ⁡  x. Then the integration becomes  =      {                            −                      1                          3                                ∫                      sin            ⁡            x                          1              +                              z                2                                      d            z                                    ,          sin          ⁡          x          ≥          0                                                  1                          3                                ∫                      sin            ⁡            x                          1              +                              z                2                                      d            z                                    ,          sin          ⁡          x          &amp;lt;          0

Integrating trig function ∫ <mrow class="MJX-TeXAtom-ORD"> <msqrt>

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