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 [latex]u[/latex] 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
[latex]\displaystyle\int f\left[g(x)\right]{g}^{\prime }(x)dx=\displaystyle\int f(u)du=F(u)+C=F(g(x))+C[/latex]

Substitution with Definite Integrals
[latex]{\displaystyle\int }_{a}^{b}f(g(x))g\text{‘}(x)dx={\displaystyle\int }_{g(a)}^{g(b)}f(u)du[/latex]

Glossary

change of variables

the substitution of a variable, such as [latex]u[/latex], 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