AP Calculus AB — calculusStandard1 min read

U-Substitution

Apply u-substitution for integration in AP Calculus AB. Handle both indefinite and definite integrals.

Tutor AI Team2026-02-25T02:13:54.044Z

U-substitution reverses the chain rule for integration. It's the most important integration technique for AP Calculus AB.

Method

Choose $u$ (the inner function).
Find $du = u'\,dx$ .
Rewrite the integral in terms of $u$ .
Integrate.
Substitute back.

For Definite Integrals

Option A: Change the limits to $u$ -limits. Option B: Substitute back and use original limits.

Worked Example: Example 1

Problem

$\int 2x(x^2+1)^3\,dx$ . Let $u = x^2+1$ , $du = 2x\,dx$ . $= \int u^3\,du = \frac{u^4}{4} + C = \frac{(x^2+1)^4}{4} + C$ .

Solution

Worked Example: Definite

Problem

$\int_0^1 xe^{x^2}\,dx$ . $u = x^2$ , $du = 2x\,dx$ → $x\,dx = \frac{du}{2}$ . Limits: $x=0 \to u=0$ , $x=1 \to u=1$ . $= \frac{1}{2}\int_0^1 e^u\,du = \frac{1}{2}(e-1)$ .

Solution

Worked Example: Example 3

Problem

$\int \frac{\cos x}{\sin x}\,dx = \int \frac{du}{u} = \ln|u| + C = \ln|\sin x| + C$ .

Solution

Practice Problems

1. $\int (3x+1)^5\,dx$ .
1. $\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx$ .
1. $\int_0^{\pi/2} \sin^3 x \cos x\,dx$ .

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Key Takeaways

✓
Choose $u$ so that $du$ (or a constant multiple) appears.
✓
For definite integrals: change the limits or substitute back.
✓
U-sub reverses the chain rule.

Tags:

#ap#calculus#integration#u-substitution

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Method

For Definite Integrals

Worked Example: Example 1

Worked Example: Definite

Worked Example: Example 3

Practice Problems

Key Takeaways

Related Topics

Slope Fields

Exponential Growth and Decay Models

Area Between Curves

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