IntegralsAdvanced

Integrate x² sin(x) by Parts

Work through repeated integration by parts for the integral of x squared times sine x.

∫ x² sin(x) dx

Method: Repeated integration by parts

Step-by-step solution

1
Choose u and dv
Let u=x² and dv=sin(x)dx so that differentiating u lowers its degree.
$u=x^2,quad dv=sin(x)dx$
2
Apply integration by parts
With du=2x dx and v=-cos(x), the first application leaves an integral of x cos(x).
$-x^2cos(x)+2int xcos(x)dx$
3
Integrate the remaining product
Apply integration by parts again with u=x and dv=cos(x)dx.
$int xcos(x)dx=xsin(x)+cos(x)$
4
Combine the terms
Substitute the second result and include the constant of integration.

-x^2cos(x)+2xsin(x)+2cos(x)+C

Differentiating the final expression returns x² sin(x), which verifies the antiderivative.

Stopping after the first integration by parts leaves another product integral unresolved.

Try the tool

Learn the concept

Step-by-step solution

Let u=x² and dv=sin(x)dx so that differentiating u lowers its degree.

u=x^2,quad dv=sin(x)dx

With du=2x dx and v=-cos(x), the first application leaves an integral of x cos(x).

-x^2cos(x)+2int xcos(x)dx

Apply integration by parts again with u=x and dv=cos(x)dx.

int xcos(x)dx=xsin(x)+cos(x)

Substitute the second result and include the constant of integration.