Integration by Parts Math Example 1

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

Example 1

easy
Find ∫xex dx\displaystyle\int x e^x\,dx.

Solution

  1. 1
    LIATE: u=xu = x, dv=ex dxdv = e^x\,dx; then du=dxdu = dx, v=exv = e^x.
  2. 2
    ∫xex dx=xex−∫ex dx=xex−ex+C\int xe^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C.
  3. 3
    Factor: ex(x−1)+Ce^x(x-1) + C.

Answer

ex(x−1)+Ce^x(x-1) + C
LIATE places algebraic before exponential, so u=xu = x. One IBP step reduces the remaining integral to something immediate.

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 2 hard

Find [formula].

Example 3 medium

Evaluate [formula] using integration by parts twice.

Example 4 easy

Find [formula].

Example 5 medium

Find [formula].

← All Integration by Parts Examples Integration by Parts Concept

Related Concepts

IntegralDerivative