Integrating Using Substitution - AP Calculus AB Study Guide

The Core Idea: Integrating Using Substitution

The method of integration using substitution, often called "u-substitution," is the primary technique for finding the antiderivatives of composite functions. While basic integration rules allow us to find antiderivatives of functions like $x^n$, $\sin (x)$, or $e^x$, they are insufficient for more complex functions like $(2x+1{)}^5$ or $x^2\cos (x^3)$. These are composite functions, where one function is nested inside another.

The entire method is based on reversing the chain rule for differentiation. The chain rule tells us how to differentiate a composite function, and substitution provides the corresponding method to integrate one. The goal is to transform a complicated integral in terms of one variable (e.g., $x$) into a much simpler integral in terms of a new variable (e.g., $u$), which can then be solved using basic antiderivative rules.

The Substitution Process

The method of substitution follows a systematic process to simplify and solve integrals of composite functions. The process is the direct reversal of the chain rule.

Let the integral be of the form $\int f(g(x))g^{\prime }(x)dx$.

1. Identify the Inner Function: Look for a composite function. Let the "inner" function be $u$. A good choice for $u$ is a function whose derivative also appears in the integrand.

   • Let $u=g(x)$.

2. Find the Differential of $u$: Differentiate $u$ with respect to $x$ to find $du$.

   • $\frac{du}{dx}=g^{\prime }(x)$, which can be written in differential form as $du=g^{\prime }(x)dx$.

3. Substitute: Replace both $g(x)$ with $u$ and $g^{\prime }(x)dx$ with $du$ in the original integral. The goal is to create a new integral entirely in terms of $u$.

   • The integral becomes $\int f(u)du$.

4. Integrate: Find the antiderivative of the new, simpler function with respect to $u$.

   • $\int f(u)du=F(u)+C$.

5. Back-Substitute: Replace $u$ with the original expression $g(x)$ to express the final answer in terms of the original variable, $x$.

   • The final answer is $F(g(x))+C$.

Understanding the Link to the Chain Rule

The method of substitution is not a new rule but rather a technique for recognizing and reversing the chain rule. The validity of the method comes directly from the chain rule for differentiation.

Recall the chain rule:

$$\frac{d}{dx}[F(g(x))]=F^{\prime }(g(x))\cdot g^{\prime }(x)$$

If we let $f(x)=F^{\prime }(x)$, then the chain rule can be written as:

$$\frac{d}{dx}[F(g(x))]=f(g(x))\cdot g^{\prime }(x)$$

By the definition of an antiderivative, if we integrate both sides, we must get back to the original function (plus a constant of integration):

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

The substitution process is a formal way to identify this pattern. By setting $u=g(x)$ and $du=g^{\prime }(x)dx$, we transform the complex integral $\int f(g(x))g^{\prime }(x)dx$ into the much simpler form $\int f(u)du$. Integrating this gives $F(u)+C$, and substituting $g(x)$ back for $u$ yields the correct result, $F(g(x))+C$.

Core Concepts & Rules

• Reversing the Chain Rule: The method of substitution is the integration counterpart to the chain rule for differentiation. It is used to find antiderivatives of composite functions.

• Composite Function Recognition: The key to substitution is identifying a composite function structure within the integrand, such as a function of a linear function (e.g., $e^{3x-1}$) or other composite functions (e.g., $\sin (x^2)$).

• The $u$ and $du$ Pair: A successful substitution requires identifying an "inner function," $u=g(x)$, such that its differential, $du=g^{\prime }(x)dx$, also appears in the integrand (perhaps missing only a constant multiplier).

• Complete Transformation: The entire integral, including the differential $dx$, must be converted from the original variable ($x$) to the new variable ($u$). No terms from the original variable should remain.

• Definite Integrals: When using substitution for a definite integral, the limits of integration, which are values of $x$, must also be converted to corresponding values of $u$.

Step-by-Step Example 1: Composite Function with a Linear Inner Function

Find the indefinite integral $\int \sqrt{5x+3}dx$.

Step 1: Identify the Inner Function

The integrand is a composite function. The "inner" function is the linear function inside the square root.

Let $u=5x+3$.

Step 2: Find the Differential of $u$

Differentiate $u$ with respect to $x$.

$\frac{du}{dx}=5$.

In differential form, this is $du=5dx$.

Step 3: Substitute

Our integral has $\sqrt{5x+3}$, which becomes $\sqrt{u}$. It also has $dx$. From Step 2, we can solve for $dx$ to get $dx=\frac{1}{5}du$. Now substitute both parts into the integral.

$$\int \sqrt{u}\cdot \left(\frac{1}{5}du\right)$$

We can pull the constant $\frac{1}{5}$ outside the integral.

$$\frac{1}{5}\int \sqrt{u}du=\frac{1}{5}\int u^{1/2}du$$

Step 4: Integrate

Now we have a simple integral that can be solved using the power rule for integration.

$$\frac{1}{5}\left(\frac{u^{1/2+1}}{1/2+1}\right)+C=\frac{1}{5}\left(\frac{u^{3/2}}{3/2}\right)+C$$

$$=\frac{1}{5}\cdot \frac{2}{3}{u}^{3/2}+C=\frac{2}{15}{u}^{3/2}+C$$

Step 5: Back-Substitute

Replace $u$ with the original expression $5x+3$.

$$\frac{2}{15}(5x+3{)}^{3/2}+C$$

Step-by-Step Example 2: Definite Integral with Substitution

Evaluate the definite integral ${\int }_0^2\frac{x}{(x^2+1{)}^2}dx$.

Step 1: Identify the Inner Function

The denominator contains the composite function. The inner function is $x^2+1$.

Let $u=x^2+1$.

Step 2: Find the Differential of $u$

Differentiate $u$ with respect to $x$.

$\frac{du}{dx}=2x$.

In differential form, this is $du=2xdx$. Our integral contains $xdx$, so we can solve for it: $xdx=\frac{1}{2}du$.

Step 3: Change the Limits of Integration

Since this is a definite integral, the limits $x=0$ and $x=2$ must be converted to $u$-values.

• Lower limit: When $x=0$, $u=(0{)}^2+1=1$.

• Upper limit: When $x=2$, $u=(2{)}^2+1=5$.

Step 4: Substitute

Rewrite the entire integral in terms of $u$, including the new limits.

The term $(x^2+1{)}^2$ becomes $u^2$. The term $xdx$ becomes $\frac{1}{2}du$.

$${\int }_1^5\frac{1}{u^2}\left(\frac{1}{2}du\right)=\frac{1}{2}{\int }_1^5{u}^{-2}du$$

Step 5: Integrate and Evaluate

Use the power rule to find the antiderivative and evaluate it at the new limits. There is no need to back-substitute to $x$ because we have already changed the limits.

$$\frac{1}{2}{\left[\frac{u^{-1}}{-1}\right]}_1^5=-\frac{1}{2}{\left[\frac{1}{u}\right]}_1^5$$

$$=-\frac{1}{2}\left(\frac{1}{5}-\frac{1}{1}\right)=-\frac{1}{2}\left(\frac{1}{5}-\frac{5}{5}\right)=-\frac{1}{2}\left(-\frac{4}{5}\right)=\frac{4}{10}=\frac{2}{5}$$

The value of the definite integral is $\frac{2}{5}$.

Using Your Calculator

The method of substitution is an analytical technique for finding indefinite integrals (antiderivatives), a process which a graphing calculator cannot perform symbolically. You cannot use a calculator to find $\frac{2}{15}(5x+3{)}^{3/2}+C$.

However, for definite integrals, the calculator is an excellent tool for checking your answer. After you have found the value of a definite integral by hand using substitution, you can verify it using your calculator's numerical integration feature.

For Example 2, ${\int }_0^2\frac{x}{(x^2+1{)}^2}dx$, you would use the following command (on a TI-84 style calculator):

fnInt(X/(X^2+1)^2, X, 0, 2)

The calculator will return $0.4$, which is the decimal equivalent of the exact answer $\frac{2}{5}$ we found by hand.

AP Exam Quick Hit

Common Question Types

• Direct Indefinite Integral: A multiple-choice question asking for the antiderivative of a function that requires substitution.

  • Example: Find $\int x^3\sqrt{x^4+5}dx$.

• Definite Integral Evaluation: A multiple-choice or free-response question that requires using substitution to find the exact value of a definite integral.

  • Example: Evaluate ${\int }_0^{\pi /4}{\sec }^2(x)e^{\tan (x)}dx$.

• Changing the Limits of Integration: A multiple-choice question that provides an integral and a substitution, and asks for the equivalent integral in terms of $u$ with the correct new limits.

  • Example: "Using the substitution $u=3x+1$, the integral ${\int }_0^1(3x+1{)}^{4}dx$ is equivalent to which of the following?" The correct answer would be $\frac{1}{3}{\int }_1^4{u}^{4}du$.

Common Mistakes

• Forgetting "+ C": When finding an indefinite integral, the constant of integration $+C$ is required for a full-credit answer.

• Incorrectly Handling Constants: When $du=k\cdot g^{\prime }(x)dx$ for some constant $k$, students often forget to include the $\frac{1}{k}$ factor in the transformed integral. For example, in $\int \cos (2x)dx$, letting $u=2x$ gives $du=2dx$, so the integral becomes $\frac{1}{2}\int \cos (u)du$. Forgetting the $\frac{1}{2}$ is a very common error.

• Mixing Variables: The new integral must be written entirely in terms of $u$ and $du$. An incorrect intermediate step like $\int \frac{u}{x}du$ cannot be integrated. Every instance of the original variable must be substituted for.

• Forgetting to Change the Bounds: For definite integrals, a frequent mistake is to substitute for $u$ but use the original $x$-bounds for evaluation. You must either change the limits to be in terms of $u$ or integrate, substitute back to $x$, and then use the original $x$-limits.

• Choosing the Wrong $u$: While not always a "mistake," choosing a non-productive $u$ can make the problem impossible to solve. The choice for $u$ should be an "inner function" whose derivative is also present in the integrand.