Integration by Substitution (Section 7.2) – Business Calculus Study Notes

Integration by Substitution

Introduction to Substitution

Integration by substitution is a fundamental technique in calculus, used to simplify the process of finding antiderivatives. It is closely related to the Chain Rule for derivatives and is essential for solving integrals involving composite functions.

• Chain Rule (for derivatives): If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

• Substitution (for integrals): Allows us to reverse the chain rule and integrate more complex expressions.

Substitution Method

Steps for Integration by Substitution

To use substitution, follow these steps:

1. Let $u$ be a suitable expression inside the integral (often the inner function).

2. Compute $du$, the differential of $u$ with respect to $x$.

3. Rewrite the integral in terms of $u$ and $du$.

4. Integrate with respect to $u$.

5. Substitute back the original variable $x$ at the end.

• Key Point: The substitution should simplify the integral, often by making the integrand a basic function of $u$.

• Example: For $\int 2x \cos(x^2) dx$, let $u = x^2$, then $du = 2x dx$.

Common Substitution Integrals

Standard Forms and Their Antiderivatives

Many integrals can be solved using substitution, especially those involving composite functions. Below is a table summarizing some common forms:

Form of the Integrand	Result
$\int f'(u) f(u) dx$	$\int f'(u) du = f(u) + c$
$\int f'(x) \cos(f(x)) dx$	$\sin(f(x)) + c$
$\int f'(x) e^{f(x)} dx$	$e^{f(x)} + c$
$\int f'(x) \frac{1}{f(x)} dx$	$\ln|f(x)| + c$

Worked Examples

Example 1: $\int 2x \cos(x^2) dx$

• Substitution: Let $u = x^2$, so $du = 2x dx$.

• Rewrite: $\int \cos(u) du$

• Integrate: $\sin(u) + c$

• Back-substitute: $\sin(x^2) + c$

Example 2: $\int (4x + 13)^3 dx$

• Substitution: Let $u = 4x + 13$, so $du = 4 dx$.

• Rewrite: $\int u^3 \frac{du}{4}$

• Integrate: $\frac{1}{4} \cdot \frac{u^4}{4} + c = \frac{u^4}{16} + c$

• Back-substitute: $\frac{(4x + 13)^4}{16} + c$

Example 3: $\int x e^{x^2} dx$

• Substitution: Let $u = x^2$, so $du = 2x dx$.

• Rewrite: $\int e^{u} \frac{du}{2}$

• Integrate: $\frac{1}{2} e^{u} + c$

• Back-substitute: $\frac{1}{2} e^{x^2} + c$

Example 4: $\int \frac{1}{x^2 + 1} dx$

• Recognize: This is a standard integral.

• Integrate: $\arctan(x) + c$

Application Example

Business Application: Debt Accumulation

Suppose a company incurs debt at a rate of $D'(t) = 1000 + 0.1t$ (dollars per year), where $t$ is the number of years since the company started. To find the total debt after $t$ years, integrate the rate function:

• Integral: $D(t) = \int (1000 + 0.1t) dt$

• Solution: $D(t) = 1000t + 0.05t^2 + c$

• Interpretation: This gives the total accumulated debt as a function of time.

Summary Table: Substitution Steps

Step	Description
1	Choose $u$ (the inner function)
2	Compute $du$
3	Rewrite the integral in terms of $u$ and $du$
4	Integrate with respect to $u$
5	Substitute back the original variable

Key Points to Remember

• Integration by substitution is the reverse process of the chain rule.

• Always check that $du$ matches the remaining part of the integrand after substitution.

• Substitution is especially useful for integrals involving composite functions.

• After integrating, always substitute back to the original variable unless otherwise specified.