U-Substitution: The Secret to Reversing the Chain Rule

U-Substitution simplifies an integral by replacing a complex 'inner' expression with a single variable $u$. It is the primary tool for reversing the Chain Rule.

Core Theorem

u = g(x)

, then

du = g'(x)dx

\int f(g(x))g'(x) \, dx = \int f(u) \, du

Practice Exercises

Example 01Easy

Evaluate

\int 2x(x^2 + 5)^3 \, dx

NEED A HINT?

Notice that

2x

is the derivative of

x^2 + 5

SHOW DETAILED EXPLANATION

Step 1: Pick $u$

Let

u = x^2 + 5

Step 2: Find $du$

du = 2x dx

Step 3: Integrate

\int u^3 \, du = \frac{1}{4}u^4 + C = \frac{1}{4}(x^2+5)^4 + C

Example 02AB/BC Standard

Evaluate

\int_0^1 x e^{x^2} \, dx

NEED A HINT?

Don't forget to change the integration bounds when moving from

x

u

SHOW DETAILED EXPLANATION

Step 1: Pick $u$ and $du$

u = x^2 \implies du = 2x dx

, so

x dx = \frac{1}{2}du

Step 2: Change Bounds

x=0, u=0^2=0

. If

x=1, u=1^2=1

Step 3: Solve

\frac{1}{2} \int_0^1 e^u \, du = \frac{1}{2}[e^u]_0^1 = \frac{1}{2}(e - 1)

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