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

The integral of x^2sin(xy) is -xcos(xy)+C

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integral of x^2sin(xy)

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

- x cos (x y) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = x y

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

Simplify

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

= - x cos (x y)

Add a constant to the solution

= - x cos (x y) + C