What is the integral of x^2ycos(xy)+2xsin(xy) ?

The integral of x^2ycos(xy)+2xsin(xy) is x^2sin(yx)+C

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integral of x^2ycos(xy)+2xsin(xy)

\int x^{2} y cos (x y) + 2 x sin (x y) d x

x^{2} sin (y x) + C

Solution steps

\int x^{2} y cos (x y) + 2 x 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 2 sin (y x) x d x + \int cos (y x) y x^{2} d x

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

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

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

Simplify

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

= x^{2} sin (y x)

Add a constant to the solution

= x^{2} sin (y x) + C

What is the integral of x^2ycos(xy)+2xsin(xy) ?
The integral of x^2ycos(xy)+2xsin(xy) is x^2sin(yx)+C