Are the sums equal to each other?They are 2 different results for the integral ∫ x e 2 x sin ⁡ ( x 3 ) d x1. − 3 1369 e 2 x ( 3 ( 35 − 74 x ) sin ⁡ ( x 3 ) + ( 37 x − 36 ) cos ⁡ ( x 3 ) ) 2. 3 37 e 2 x ( x ( 6 sin ⁡ ( x 3 ) − cos ⁡ ( x 3 ) ) − 3 37 ( 35 sin ⁡ ( x 3 ) − 13 cos ⁡ ( x 3 ) ) ) Are the sums equal to each other?

Question

Are the sums equal to each other?They are   2 different results for the integral  ∫  x      e          2      x        sin  ⁡      (          x      3        )      d  x1.                     −        3            1369              e              2        x                    (      3      (      35      −      74      x      )      sin      ⁡              (                  x          3                )            +      (      37      x      −      36      )      cos      ⁡              (                  x          3                )            )      2.             3      37              e              2        x                    (      x              (        6        sin        ⁡                  (                      x            3                    )                −        cos        ⁡                  (                      x            3                    )                )            −              3        37                    (        35        sin        ⁡                  (                      x            3                    )                −        13        cos        ⁡                  (                      x            3                    )                )            )      Are the sums equal to each other?

Stevinivm · Accepted Answer

Second equation:      3    1369        e          2      x        (  37  (  6  x  sin  ⁡  (  x      /    3  )  −  x  cos  ⁡  (  x      /    3  )  )  −  3  (  35  sin  ⁡  (  x      /    3  )  −  12  cos  ⁡  (  x      /    3  )  )  =      3    1369        e          2      x        (  222  x  sin  ⁡  (  x      /    3  )  −  37  x  cos  ⁡  (  x      /    3  )  −  105  sin  ⁡  (  x      /    3  )  +  36  cos  ⁡  (  x      /    3  )  )  =      3    1369        e          2      x        (  3  (  74  x  −  34  )  sin  ⁡  (  x      /    3  )  −  37  x  cos  ⁡  (  x      /    3  )  +  36  cos  ⁡  (  x      /    3  )  )  =            −      3        1369        e          2      x        (  3  (  34  −  74  x  )  sin  ⁡  (  x      /    3  )  +  (  37  x  −  36  )  cos  ⁡  (  x      /    3  )  )So they are equal.

Makayla Boyd · Accepted Answer

Set them equal to each other and divide both sides by e^(2x).  −            3              (        (        −        36        +        37        x        )        cos        ⁡                  (                      x            3                    )                +        3        (        35        −        74        x        )        sin        ⁡                  (                      x            3                    )                )              1369    =      3    37        (    x          (      −      cos      ⁡              (                  x          3                )            +      6      sin      ⁡              (                  x          3                )            )        −          3      37              (      −      12      cos      ⁡              (                  x          3                )            +      35      sin      ⁡              (                  x          3                )            )        )  Multiply both sides by 1369.  −  3      (    (    −    36    +    37    x    )    cos    ⁡          (              x        3            )        +    3    (    35    −    74    x    )    sin    ⁡          (              x        3            )        )    =  111      (    x          (      −      cos      ⁡              (                  x          3                )            +      6      sin      ⁡              (                  x          3                )            )        −          3      37              (      −      12      cos      ⁡              (                  x          3                )            +      35      sin      ⁡              (                  x          3                )            )        )  Divide both sides by 3.  −  (  −  36  +  37  x  )  cos  ⁡      (          x      3        )    −  3  (  35  −  74  x  )  sin  ⁡      (          x      3        )    =  37      (    x          (      −      cos      ⁡              (                  x          3                )            +      6      sin      ⁡              (                  x          3                )            )        −          3      37              (      −      12      cos      ⁡              (                  x          3                )            +      35      sin      ⁡              (                  x          3                )            )        )  Start cleaning up.  (  36  −  37  x  )  cos  ⁡      (          x      3        )    −  (  105  −  222  x  )  sin  ⁡      (          x      3        )    =  37      (    x          (      −      cos      ⁡              (                  x          3                )            +      6      sin      ⁡              (                  x          3                )            )        −          3      37              (      −      12      cos      ⁡              (                  x          3                )            +      35      sin      ⁡              (                  x          3                )            )        )    (  36  −  37  x  )  cos  ⁡      (          x      3        )    −  (  105  −  222  x  )  sin  ⁡      (          x      3        )    =  37      (    −    x    cos    ⁡          (              x        3            )        +    6    x    sin    ⁡          (              x        3            )        −          3      37              (      −      12      cos      ⁡              (                  x          3                )            +      35      sin      ⁡              (                  x          3                )            )        )    (  36  −  37  x  )  cos  ⁡      (          x      3        )    −  (  105  −  222  x  )  sin  ⁡      (          x      3        )    =  −  37  x  cos  ⁡      (          x      3        )    +  222  x  sin  ⁡      (          x      3        )    −  3      (    −    12    cos    ⁡          (              x        3            )        +    35    sin    ⁡          (              x        3            )        )    (  36  −  37  x  )  cos  ⁡      (          x      3        )    −  (  105  −  222  x  )  sin  ⁡      (          x      3        )    =  36  cos  ⁡      (          x      3        )    −  37  x  cos  ⁡      (          x      3        )    −  105  sin  ⁡      (          x      3        )    +  222  x  sin  ⁡      (          x      3        )    36  cos  ⁡      (          x      3        )    −  37  x  cos  ⁡      (          x      3        )    −  (  105  −  222  x  )  sin  ⁡      (          x      3        )    =  36  cos  ⁡      (          x      3        )    −  37  x  cos  ⁡      (          x      3        )    −  105  sin  ⁡      (          x      3        )    +  222  x  sin  ⁡      (          x      3        )  Easy to see now that they are equal.

Are the sums equal to each other? They are 2 different results for the integral &#x222B

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