What is the integral of (xsin(xy)) ?

The integral of (xsin(xy)) is -1/y xcos(yx)+1/(y^2)sin(yx)+C

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integral of (xsin(xy))

\int (x sin (x y)) d x

- \frac{1}{y} x cos (y x) + \frac{1}{y ^{2}} sin (y x) + C

Solution steps

\int x sin (x y) d x

Apply Integration By Parts

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

\int - \frac{1}{y} cos (y x) d x = - \frac{1}{y ^{2}} sin (y x)

= - \frac{1}{y} x cos (y x) - (- \frac{1}{y ^{2}} sin (y x))

Simplify

= - \frac{1}{y} x cos (y x) + \frac{1}{y ^{2}} sin (y x)

Add a constant to the solution

= - \frac{1}{y} x cos (y x) + \frac{1}{y ^{2}} sin (y x) + C

What is the integral of (xsin(xy)) ?
The integral of (xsin(xy)) is -1/y xcos(yx)+1/(y^2)sin(yx)+C