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U-Substitution

Unit 6 · Integration & Accumulation

U-Substitution

U-substitution reverses the Chain Rule: let u = g(x) so du = g′(x) dx, rewrite the integral entirely in u, integrate, then substitute back. For definite integrals, also change the limits of integration when you substitute.

Substitution Steps

1. Choose u = inner function
2. Compute du = g′(x) dx, solve for dx
3. Rewrite integral in u; for definite integrals, change limits
4. Integrate; substitute back for indefinite case
Caution: For definite integrals: change the limits to u-values after substituting — do NOT convert back to x at the end.
AP Tip: If the integrand is a rational function with numerator degree ≥ denominator degree, use long division first.
Type 1

U-Substitution — Indefinite Integrals

Identify the inner function, substitute, integrate in u, then substitute back to x.

Example 1
Evaluate the integral.
∫3x2sin⁡(x3) dx\int 3x^2 \sin(x^3)\,dx
Example 2
Evaluate the integral.
∫cos⁡(ln⁡x)x dx\int \frac{\cos(\ln x)}{x}\,dx
Example 3
Evaluate the integral.
∫xx2+4 dx\int \frac{x}{\sqrt{x^2+4}}\,dx

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Type 2

U-Substitution — Definite Integrals (Change Limits)

After substituting u = g(x), convert the limits a → g(a) and b → g(b), then evaluate directly in u.

Example 4
Evaluate the definite integral.
∫01x ex2 dx\int_0^1 x\,e^{x^2}\,dx
Example 5
Evaluate the definite integral.
∫0π/2sin⁡3xcos⁡x dx\int_0^{\pi/2} \sin^3 x\cos x\,dx

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Type 3

Long Division Before Integrating

When the numerator's degree ≥ the denominator's degree, divide first to reduce the integrand.

Example 6
Evaluate the integral.
∫x2+3xx−1 dx\int \frac{x^2 + 3x}{x-1}\,dx

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U-Substitution | AP Calculus AB — Mathfolis