Integrating Using Substitution - AP Calculus BC Study Guide

The Core Idea: Integrating Using Substitution

Integration can be thought of as the reverse process of differentiation. The chain rule allows us to differentiate composite functions, so we need an equivalent method to integrate them. The method of substitution, also known as u-substitution, is the corresponding technique for finding the antiderivative of a composite function. This method is specifically designed to handle integrals where the integrand is the result of a chain rule differentiation.

The fundamental goal of substitution is to transform a complicated integral involving one variable (e.g., $x$) into a simpler, more recognizable integral in terms of a new variable (e.g., $u$). This is accomplished by identifying an "inner function" within the integrand whose derivative is also present as a factor. By substituting for this inner function and its differential, we can simplify the problem, integrate, and then substitute back to find the answer in terms of the original variable.

Key Formulas/Rules

The method of substitution is based on reversing the chain rule. It is applied to indefinite integrals that fit the specific structure below.

The core transformation is:

$$\int f(g(x))g^{\prime }(x)dx$$

To evaluate this integral, we perform the following steps:

1. Define $u$: Let $u$ be the "inner function," $g(x)$.

   $$u=g(x)$$

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

   $$\frac{du}{dx}=g^{\prime }(x)⟹du=g^{\prime }(x)dx$$

3. Substitute: Replace both $g(x)$ with $u$ and $g^{\prime }(x)dx$ with $du$ in the original integral.

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

4. Integrate: Evaluate the new, simpler integral with respect to $u$. Let $F$ be the antiderivative of $f$.

   $$\int f(u)du=F(u)+C$$

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

   $$F(g(x))+C$$

Understanding the Connection to the Chain Rule

The method of substitution is not a new, arbitrary rule; it is a direct consequence of reversing the chain rule for differentiation. Understanding this connection is key to mastering the concept.

Recall the chain rule for differentiation:

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

Let's define a new function $f$ such that $f(u)=F^{\prime }(u)$. Substituting this into the chain rule gives:

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

Now, consider the definition of an indefinite integral (or antiderivative). If we integrate the right side of the equation above, we must get the function inside the derivative on the left side, plus a constant of integration:

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

This is precisely the result we get from the method of substitution. The substitution $u=g(x)$ transforms the integral into $\int f(u)du$, whose antiderivative is $F(u)+C$. Substituting $g(x)$ back in for $u$ yields the final answer, $F(g(x))+C$, confirming that the method is a systematic way of reversing the chain rule.

Core Concepts & Rules

• Reversing the Chain Rule: Integration by substitution is the technique used to find the antiderivative of functions that are the result of a chain rule differentiation.

• Pattern Recognition: This method is applicable when the integrand contains a composite function, $f(g(x))$, and the derivative of the inner function, $g^{\prime }(x)$, is also present as a multiplicative factor.

• The Substitution Process: The core of the method is to define a new variable, $u$, equal to the inner function $g(x)$. The differential $du$ is then found by differentiating $u$ ($du=g^{\prime }(x)dx$).

• Complete Transformation: The goal is to rewrite the entire integral in terms of $u$ and $du$. No terms involving the original variable $x$ should remain after the substitution.

• Back-Substitution: After finding the antiderivative in terms of $u$, the final step is always to substitute the original expression $g(x)$ back in for $u$, so the final answer is a function of $x$.

• Constant of Integration: Because the method is used for indefinite integrals, the final answer must always include the constant of integration, + C.

Step-by-Step Example 1: Basic Application

Problem: Evaluate the indefinite integral $\int 3x^2(x^3+5{)}^{7}dx$.

This integral fits the form $\int f(g(x))g^{\prime }(x)dx$. The inner function is $g(x)=x^3+5$, and its derivative, $g^{\prime }(x)=3x^2$, is also present as a factor.

Step 1: Identify and define $u$.

Let $u$ be the inner function.

$$u=x^3+5$$

Step 2: Find the differential $du$.

Differentiate $u$ with respect to $x$.

$$\frac{du}{dx}=3x^2$$

Solve for $du$.

$$du=3x^{2}dx$$

Step 3: Substitute $u$ and $du$ into the integral.

Replace $(x^3+5)$ with $u$ and $3x^{2}dx$ with $du$.

$$\int (x^3+5{)}^7(3x^{2}dx)=\int u^{7}du$$

Step 4: Integrate with respect to $u$.

Use the power rule for integration.

$$\int u^{7}du=\frac{u^8}{8}+C$$

Step 5: Substitute back to express the answer in terms of $x$.

Replace $u$ with $x^3+5$.

$$\frac{(x^3+5{)}^8}{8}+C$$

Final Answer:

$$\int 3x^2(x^3+5{)}^{7}dx=\frac{(x^3+5{)}^8}{8}+C$$

Step-by-Step Example 2: Exam-Style Application

Problem: Find the antiderivative of $\int \frac{\cos (x)}{e^{\sin (x)}}dx$.

This problem requires recognizing the inner function and adjusting for a constant if necessary. First, rewrite the integrand to make the structure clearer.

$$\int \cos (x)e^{-\sin (x)}dx$$

The inner function appears to be in the exponent, $g(x)=-\sin (x)$. Its derivative is $g^{\prime }(x)=-\cos (x)$. The integrand has $\cos (x)$, which is off by a factor of -1.

Step 1: Identify and define $u$.

Let $u$ be the inner function in the exponent.

$$u=-\sin (x)$$

Step 2: Find the differential $du$.

Differentiate $u$ with respect to $x$.

$$\frac{du}{dx}=-\cos (x)$$

Solve for $du$.

$$du=-\cos (x)dx$$

Step 3: Adjust for the constant and substitute.

Our integral has $\cos (x)dx$, not $-\cos (x)dx$. We can solve our differential expression for $\cos (x)dx$:

$$-du=\cos (x)dx$$

Now substitute $u$ for $-\sin (x)$ and $-du$ for $\cos (x)dx$.

$$\int e^{-\sin (x)}(\cos (x)dx)=\int e^u(-du)$$

Step 4: Simplify and integrate with respect to $u$.

Factor out the constant (-1).

$$-\int e^{u}du=-e^u+C$$

Step 5: Substitute back to express the answer in terms of $x$.

Replace $u$ with $-\sin (x)$.

$$-e^{-\sin (x)}+C$$

Final Answer:

$$\int \frac{\cos (x)}{e^{\sin (x)}}dx=-e^{-\sin (x)}+C=-\frac{1}{e^{\sin (x)}}+C$$

Using Your Calculator

A graphing calculator cannot find an indefinite integral symbolically (i.e., it cannot produce the expression $F(x)+C$). The method of substitution is a purely analytical technique that must be performed by hand.

However, a calculator is an excellent tool for checking your answer.

Suppose you found that $\int f(x)dx=F(x)+C$. To verify this, you can use the fact that the derivative of your answer $F(x)$ should be the original integrand $f(x)$.

Steps to Check Your Answer:

1. Perform the integration by hand: Find the antiderivative $F(x)$ using the method of substitution. For the example $\int 3x^2(x^3+5{)}^{7}dx$, your answer is $F(x)=\frac{1}{8}(x^3+5{)}^8$.

2. Use the numerical derivative function: On your calculator, use the numerical derivative feature (e.g., nDeriv on a TI-84 or $d/dx(...)$ from the math menu) to find the derivative of your answer, $F(x)$.

3. Compare with the integrand:

   • Graphical Check: In Y1, enter the original integrand, f(x) = 3x^2(x^3 + 5)^7. In Y2, enter the numerical derivative of your answer, e.g., nDeriv( (1/8)(X^3+5)^8, X, X). If your answer is correct, the graphs of Y1 and Y2 should be identical.

   • Numerical Check: Evaluate the original integrand $f(x)$ at a specific value, say $x=1$. Then, evaluate the numerical derivative of your answer $F(x)$ at $x=1$. The two values should be equal.

AP Exam Quick Hit

Common Question Types

• Direct Antiderivative (Multiple Choice): A question will ask for the evaluation of an indefinite integral where substitution is the required method.

  • Example: Find $\int \frac{x}{\sqrt{1-4x^2}}dx$. (Requires $u=1-4x^2$).

• Finding a Particular Solution (Free Response): A question provides a differential equation $dy/dx$ and an initial condition. The first step in solving is to integrate $dy/dx$, which often requires substitution.

  • Example: A particle's velocity is given by $v(t)=t\cdot \cos (t^2)$. Find the position function $s(t)$ if $s(0)=5$. (Requires integrating $v(t)$ using $u=t^2$).

• Trigonometric Integrals (Multiple Choice/Free Response): Integrals that involve combinations of trigonometric functions are very common and frequently require substitution.

  • Example: Evaluate $\int {\sec }^2(3x)e^{\tan (3x)}dx$. (Requires $u=\tan (3x)$).

Common Mistakes

• Forgetting the Constant of Integration: The most frequent error on indefinite integral problems is omitting the + C in the final answer.

• Incorrectly Handling Constants: When $du$ is a constant multiple of the term in the integrand (e.g., $du=2xdx$ but you only have $xdx$), students often forget to introduce the reciprocal of that constant (e.g., $1/2$) outside the new integral.

• Mixing Variables $x$ and $u$: After substituting, the new integral must be entirely in terms of $u$ and $du$. An incorrect intermediate step like $\int u^{7}xdx$ is a common mistake and cannot be integrated.

• Forgetting to Substitute Back: The final answer must be a function of the original variable ($x$). A common error is to correctly integrate with respect to $u$ but then fail to substitute $g(x)$ back into the final expression.

• Choosing the Wrong $u$: If the derivative of your chosen $u$ is not present as a factor in the integrand, the substitution will not work. For example, in $\int x\sin (x^2)dx$, choosing $u=\sin (x^2)$ is incorrect because its derivative is $\cos (x^2)\cdot 2x$, and $\cos (x^2)$ is not in the integrand. The correct choice is $u=x^2$.