What is the integral of xy^2-cos(x)sin(x) ?

The integral of xy^2-cos(x)sin(x) is -(sin^2(x))/2+(y^2x^2)/2+C

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integral of xy^2-cos(x)sin(x)

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

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

Solution steps

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

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

\int y^{2} x d x = \frac{y ^{2} x ^{2}}{2}

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

Add a constant to the solution

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

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What is the integral of xy^2-cos(x)sin(x) ?
The integral of xy^2-cos(x)sin(x) is -(sin^2(x))/2+(y^2x^2)/2+C