---
title: "Integration by Parts"
description: "Reverse the product rule to trade one integral for another that is easier, and choose the parts so the trade is an improvement."
canonical: https://lightmysky.com/learn/mathematics/integration-by-parts-mt_v6jZVs2_tf
source: https://lightmysky.com/learn/mathematics/integration-by-parts-mt_v6jZVs2_tf.md
retrieved: 2026-09-12
---

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# Integration by Parts

Reverse the product rule to trade one integral for another that is easier, and choose the parts so the trade is an improvement.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/integration-by-parts-mt_v6jZVs2_tf

## Ready when they can

- Choose u and dv so the new integral is simpler than the original
- Apply the method twice and solve for the original integral when it reappears
- Integrate ln x by taking dv = dx

## Lesson: Trade one integral for an easier one

Some products cannot be split by substitution, like x sin x. Read the product rule backwards instead: the integral of u dv equals u v minus the integral of v du. You trade one integral for another, so choose the trade that gets simpler.

**Example.** Take the integral of x e^x dx. LIATE puts algebra before exponentials, so set u = x and dv = e^x dx. Then du = dx and v = e^x. You get x e^x minus the integral of e^x dx, which is x e^x - e^x + C. The reverse split would give a harder integral, so the choice matters.

A lone log is welcome too: for ln x, set u = ln x and dv = dx. Sometimes the old integral returns, as with e^x cos x, and then you collect it on the left and solve for it. A square like x^2 e^x simply needs two full rounds.

**Tip.** Pick u by LIATE: logs, then inverse trig, then algebra, with trig and exponentials as dv. Whatever you call dv must be something you can integrate. Check your answer by differentiating, since a sign slip on the minus term is the classic error.

**Recap.** Split into u and dv with LIATE, subtract the new integral, and solve when the old one returns.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Integration by Substitution](https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7)
- [The Product and Quotient Rules](https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv)

## Opens up

- [Weak Solutions and Test Functions](https://lightmysky.com/learn/mathematics/weak-solutions-and-test-functions-mt_AKlC3Ol70p)
- [Trigonometric Integrals and Trigonometric Substitution](https://lightmysky.com/learn/mathematics/trigonometric-integrals-and-trigonometric-substitution-mt_JqysS41psg)