Integration by Parts Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

easy
Find ∫xcos⁡x dx\displaystyle\int x\cos x\,dx.

Solution

  1. 1
    u=xu = x, dv=cos⁡x dxdv = \cos x\,dx; du=dxdu = dx, v=sin⁡xv = \sin x.
  2. 2
    xsin⁡x−∫sin⁡x dx=xsin⁡x+cos⁡x+Cx\sin x - \int \sin x\,dx = x\sin x + \cos x + C.

Answer

xsin⁡x+cos⁡x+Cx\sin x + \cos x + C
One IBP application with the algebraic factor as uu yields an immediately integrable result.

About Integration by Parts

An integration technique based on the product rule: ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du. Used when the integrand is a product of two functions.

Learn more about Integration by Parts →

More Integration by Parts Examples

Example 1 easy

Find [formula].

Example 2 hard

Find [formula].

Example 3 medium

Evaluate [formula] using integration by parts twice.

Example 5 medium

Find [formula].

← All Integration by Parts Examples Integration by Parts Concept

Related Concepts

IntegralDerivative