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Integration by Parts (IBP)

IBP is the reverse of the product rule for integration.

Core Theorem
∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du

TL;DR: Use LIATE to pick $u$, then apply the formula.

Practice Exercises

Example 01Medium
Evaluate ∫xex dx\int x e^x \, dx.
NEED A HINT?
Use LIATE: xx is Algebraic (choose as uu), exe^x is Exponential (choose as dvdv).
SHOW DETAILED EXPLANATION

Step 1: Assign $u$ and $dv$

u=x,dv=exdxu = x, dv = e^x dx.

Step 2: Find $du$ and $v$

du=dx,v=exdu = dx, v = e^x.

Step 3: Apply Formula

xex−∫ex dx=xex−ex+Cxe^x - \int e^x \, dx = xe^x - e^x + C.
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