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U-substitution

This original textbook reading covers part of the AP topic. Follow the AP course outline for complete topic instruction and practice.

U-substitution

A technique used for solving integrals, where a complex term is substituted for the variable u, and the bounds for integration changed accordingly.

Example:

$\int_{0}^{\sqrt{π}} 2 x \sin (x^{2}) d x$ $u = x^{2} \Rightarrow d u = 2 x d x$

$as x \to 0, u \to 0; as x \to \sqrt{π}, u \to π$

$\int_{0}^{\sqrt{π}} 2 x \sin (x^{2}) d x = \int_{0}^{π} \sin u d u = {[- \cos u]}_{0}^{π}$

$= (- \cos π) - (- \cos 0) = 1 + 1 = 2$

Related Glossary Terms

Definite integral, Indefinite integral, Trigonometric substitution

Index

Chapter 8 - FFR - Substitution and Differentials

Chapter 9 - Trigonometric Substitution

Glossary and deeper reading · 1 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Substitution for integration
Substitution rewrites an integral using a new variable whose differential accounts for a derivative factor. It reverses the chain rule for suitable compositions. For example, with u=x²+1 and du=2x dx, the integral of 2x(x²+1)³ becomes the integral of u³. For a definite integral, transform the bounds consistently or return to the original variable before evaluating them.
Read more: 6.9 · Substitution for integration →

Substitution for integration

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