Summary of Substitution | Calculus I

Module 5: Integration

Summary of Substitution

Essential Concepts

Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term ‘substitution’ refers to changing variables or substituting the variable $u$ and du for appropriate expressions in the integrand.
When using substitution for a definite integral, we also have to change the limits of integration.

Key Equations

Substitution with Indefinite Integrals
$\displaystyle\int f\left[g(x)\right]{g}^{\prime }(x)dx=\displaystyle\int f(u)du=F(u)+C=F(g(x))+C$
Substitution with Definite Integrals
${\displaystyle\int }_{a}^{b}f(g(x))g\text{‘}(x)dx={\displaystyle\int }_{g(a)}^{g(b)}f(u)du$

Glossary

change of variables: the substitution of a variable, such as $u$ , for an expression in the integrand

integration by substitution: a technique for integration that allows integration of functions that are the result of a chain-rule derivative