What is the integral of xysin(x^2+y^2) ?

The integral of xysin(x^2+y^2) is -1/2 ycos(y^2+x^2)+C

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integral of xysin(x^2+y^2)

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{sin ( u )}{2} d u

= y \cdot \int \frac{sin ( u )}{2} d u

Take the constant out:

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

= y \frac{1}{2} \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= y \frac{1}{2} (- cos (u))

Substitute back

u = y^{2} + x^{2}

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

Simplify

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

Add a constant to the solution

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

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What is the integral of xysin(x^2+y^2) ?
The integral of xysin(x^2+y^2) is -1/2 ycos(y^2+x^2)+C