### Learning Outcomes

- Recognize the conditions under which substitution may be used to evaluate integrals
- Use substitution to evaluate indefinite integrals

At first, the approach to the substitution procedure may not appear very obvious. However, it is primarily a visual task—that is, the integrand shows you what to do; it is a matter of recognizing the form of the function. So, what are we supposed to see? We are looking for an integrand of the form [latex]f\left[g(x)\right]{g}^{\prime }(x)dx.[/latex] For example, in the integral [latex]\displaystyle\int {({x}^{2}-3)}^{3}2xdx,[/latex] we have [latex]f(x)={x}^{3},g(x)={x}^{2}-3,[/latex] and [latex]g\text{‘}(x)=2x.[/latex] Then,

and we see that our integrand is in the correct form.

The method is called *substitution* because we substitute part of the integrand with the variable [latex]u[/latex] and part of the integrand with *du*. It is also referred to as **change of variables** because we are changing variables to obtain an expression that is easier to work with for applying the integration rules.

### Substitution with Indefinite Integrals

Let [latex]u=g(x),,[/latex] where [latex]{g}^{\prime }(x)[/latex] is continuous over an interval, let [latex]f(x)[/latex] be continuous over the corresponding range of [latex]g[/latex], and let [latex]F(x)[/latex] be an antiderivative of [latex]f(x).[/latex] Then,

### Proof

Let [latex]f[/latex], [latex]g[/latex], [latex]u[/latex], and *F* be as specified in the theorem. Then

Integrating both sides with respect to [latex]x[/latex], we see that

If we now substitute [latex]u=g(x),[/latex] and [latex]du=g\text{‘}(x)dx,[/latex] we get

[latex]_\blacksquare[/latex]

Returning to the problem we looked at originally, we let [latex]u={x}^{2}-3[/latex] and then [latex]du=2xdx.[/latex] Rewrite the integral in terms of [latex]u[/latex]:

Using the power rule for integrals, we have

Substitute the original expression for [latex]x[/latex] back into the solution:

We can generalize the procedure in the following Problem-Solving Strategy.

### Problem-Solving Strategy: Integration by Substitution

- Look carefully at the integrand and select an expression [latex]g(x)[/latex] within the integrand to set equal to [latex]u[/latex]. Let’s select [latex]g(x).[/latex] such that [latex]{g}^{\prime }(x)[/latex] is also part of the integrand.
- Substitute [latex]u=g(x)[/latex] and [latex]du={g}^{\prime }(x)dx.[/latex] into the integral.
- We should now be able to evaluate the integral with respect to [latex]u[/latex]. If the integral can’t be evaluated we need to go back and select a different expression to use as [latex]u[/latex].
- Evaluate the integral in terms of [latex]u[/latex].
- Write the result in terms of [latex]x[/latex] and the expression [latex]g(x).[/latex]

### Example: Evaluating an inDefinite Integral Using Substitution

Use substitution to find the antiderivative of [latex]\displaystyle\int 6x{(3{x}^{2}+4)}^{4}dx.[/latex]

Watch the following video to see the worked solution to Example: Evaluating an Indefinite Integral Using Substitution.

### Try It

Use substitution to find the antiderivative of [latex]\displaystyle\int 3{x}^{2}{({x}^{3}-3)}^{2}dx.[/latex]

Sometimes we need to adjust the constants in our integral if they don’t match up exactly with the expressions we are substituting.

Tip: As long as you select a [latex]g(x)[/latex] for [latex]u[/latex] such that *a multiple* of [latex]g'(x)[/latex] exists in the integrand, it will work! In other words, make sure the exponents work – don’t worry about the constants. For instance, in the example below, if we select [latex]{u={z}^{2}-5}[/latex], [latex]g'(x)={2z}[/latex]. Although [latex]g'(x)={2z}[/latex] doesn’t appear in the integrand, [latex]z[/latex] does. Substitution can work here! Watch how:

### Example: Using Substitution with Alteration

Use substitution to find the antiderivative of [latex]\displaystyle\int z\sqrt{{z}^{2}-5}dz.[/latex]

Watch the following video to see the worked solution to Example: Using Substitution with Alteration.

### Try It

Use substitution to find the antiderivative of [latex]\displaystyle\int {x}^{2}{({x}^{3}+5)}^{9}dx.[/latex]

### Example: Using Substitution with Integrals of Trigonometric Functions

Use substitution to evaluate the integral [latex]\displaystyle\int \frac{ \sin t}{{ \cos }^{3}t}dt.[/latex]

### Try It

Use substitution to evaluate the integral [latex]\displaystyle\int \frac{ \cos t}{{ \sin }^{2}t}dt.[/latex]

Sometimes we need to manipulate an integral in ways that are more complicated than just multiplying or dividing by a constant. We need to eliminate all the expressions within the integrand that are in terms of the original variable. When we are done, [latex]u[/latex] should be the only variable in the integrand. In some cases, this means solving for the original variable in terms of [latex]u[/latex]. This technique should become clear in the next example.

### Example: Finding an Antiderivative Using [latex]u[/latex]-Substitution

Use substitution to find the antiderivative of [latex]\displaystyle\int \frac{x}{\sqrt{x-1}}dx.[/latex]

### Try It

Use substitution to evaluate the indefinite integral [latex]\displaystyle\int { \cos }^{3}t \sin tdt.[/latex]

Let [latex]u={x}^{3}-3.[/latex]

[latex]\displaystyle\int 3{x}^{2}{({x}^{3}-3)}^{2}dx=\frac{1}{3}{({x}^{3}-3)}^{3}+C[/latex]