What is the integral of cos(xy)-xysin(xy) ?

The integral of cos(xy)-xysin(xy) is xcos(yx)+C

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integral of cos(xy)-xysin(xy)

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

x cos (y x) + C

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos (y x) d x - \int sin (y x) y x d x

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

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

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

Simplify

= x cos (y x)

Add a constant to the solution

= x cos (y x) + C

What is the integral of cos(xy)-xysin(xy) ?
The integral of cos(xy)-xysin(xy) is xcos(yx)+C