## Substitution for Definite Integrals

### Learning Outcomes

• Use substitution to evaluate definite integrals.

Substitution can be used with definite integrals, too. However, using substitution to evaluate a definite integral requires a change to the limits of integration. If we change variables in the integrand, the limits of integration change as well.

### Substitution with Definite Integrals

Let $u=g(x)$ and let ${g}^{\text{′}}$ be continuous over an interval $\left[a,b\right],$ and let $f$ be continuous over the range of $u=g(x).$ Then,

${\displaystyle\int }_{a}^{b}f(g(x)){g}^{\prime }(x)dx={\displaystyle\int }_{g(a)}^{g(b)}f(u)du$

Although we will not formally prove this theorem, we justify it with some calculations here. From the substitution rule for indefinite integrals, if $F(x)$ is an antiderivative of $f(x),$ we have

$\displaystyle\int f(g(x)){g}^{\prime }(x)dx=F(g(x))+C$

Then

$\begin{array}{cc}{\displaystyle\int }_{a}^{b}f\left[g(x)\right]{g}^{\prime }(x)dx\hfill & ={F(g(x))|}_{x=a}^{x=b}\hfill \\ & =F(g(b))-F(g(a))\hfill \\ & ={F(u)|}_{u=g(a)}^{u=g(b)}\hfill \\ \\ \\ & ={\displaystyle\int }_{g(a)}^{g(b)}f(u)du,\hfill \end{array}$

and we have the desired result.

### example: Using Substitution to Evaluate a Definite Integral

Use substitution to evaluate ${\displaystyle\int }_{0}^{1}{x}^{2}{(1+2{x}^{3})}^{5}dx.$

Watch the following video to see the worked solution to Example: Using Substitution to Evaluate a Definite Integral.

### Try It

Use substitution to evaluate the definite integral ${\displaystyle\int }_{-1}^{0}y{(2{y}^{2}-3)}^{5}dy.$

### example: Using Substitution with an Exponential Function

Use substitution to evaluate ${\displaystyle\int }_{0}^{1}x{e}^{4{x}^{2}+3}dx.$

### Try It

Use substitution to evaluate ${\displaystyle\int }_{0}^{1}{x}^{2} \cos \left(\frac{\pi }{2}{x}^{3}\right)dx.$

Substitution may be only one of the techniques needed to evaluate a definite integral. All of the properties and rules of integration apply independently, and trigonometric functions may need to be rewritten using a trigonometric identity before we can apply substitution. Also, we have the option of replacing the original expression for $u$ after we find the antiderivative, which means that we do not have to change the limits of integration. These two approaches are shown in the following examples.

### example: Using Substitution to Evaluate a Trigonometric Integral

Use substitution to evaluate ${\displaystyle\int }_{0}^{\pi \text{/}2}{ \cos }^{2}\theta d\theta .$

Watch the following video to see the worked solution to Example: Using Substitution to Evaluate a Trigonometric Integral.