What is the integral of x^3cos(xy) ?

The integral of x^3cos(xy) is x^2sin(xy)+C

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integral of x^3cos(xy)

\int x^{3} cos (x y) d y

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

Solution steps

\int x^{3} cos (x y) d y

Take the constant out:

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

= x^{3} \cdot \int cos (x y) d y

Apply u-substitution

= x^{3} \cdot \int \frac{cos ( u )}{x} d u

Take the constant out:

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

= x^{3} \frac{1}{x} \cdot \int cos (u) d u

Use the common integral:

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

= x^{3} \frac{1}{x} sin (u)

Substitute back

u = x y

= x^{3} \frac{1}{x} sin (x y)

Simplify

x^{3} \frac{1}{x} sin (x y) : x^{2} sin (x y)

= x^{2} sin (x y)

Add a constant to the solution

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