What is the integral of e^{xy}sin(xy) ?

The integral of e^{xy}sin(xy) is -(e^{yx}cos(yx))/(2y)+(e^{yx}sin(yx))/(2y)+C

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integral of e^{xy}sin(xy)

\int e^{x y} sin (x y) d x

- \frac{e ^{y x} cos ( y x )}{2 y} + \frac{e ^{y x} sin ( y x )}{2 y} + C

Solution steps

\int e^{x y} sin (x y) d x

Apply Integration By Parts

: - e^{y x} \frac{1}{y} cos (y x) - \int - e^{y x} cos (y x) d x

= - e^{y x} \frac{1}{y} cos (y x) - \int - e^{y x} cos (y x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - e^{y x} \frac{1}{y} cos (y x) - (- \int e^{y x} cos (y x) d x)

Apply Integration By Parts

: e^{y x} \frac{1}{y} sin (y x) - \int e^{y x} sin (y x) d x

= - e^{y x} \frac{1}{y} cos (y x) - (- (e^{y x} \frac{1}{y} sin (y x) - \int e^{y x} sin (y x) d x))

Therefore

\int e^{y x} sin (y x) d x = - e^{y x} \frac{1}{y} cos (y x) - (- (e^{y x} \frac{1}{y} sin (y x) - \int e^{y x} sin (y x) d x))

Isolate

\int e^{y x} sin (y x) d x

= - \frac{e ^{y x} cos ( y x )}{2 y} + \frac{e ^{y x} sin ( y x )}{2 y}

Add a constant to the solution

= - \frac{e ^{y x} cos ( y x )}{2 y} + \frac{e ^{y x} sin ( y x )}{2 y} + C

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What is the integral of e^{xy}sin(xy) ?
The integral of e^{xy}sin(xy) is -(e^{yx}cos(yx))/(2y)+(e^{yx}sin(yx))/(2y)+C