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U-Substitution: 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
If u=g(x)u = g(x), then du=g′(x)dxdu = g'(x)dx:
∫f(g(x))g′(x) dx=∫f(u) du\int f(g(x))g'(x) \, dx = \int f(u) \, du

TL;DR: Identify the 'inner' function $u$, find $du$, and substitute to simplify the integral.

Practice Exercises

Example 01Easy
Evaluate ∫2x(x2+5)3 dx\int 2x(x^2 + 5)^3 \, dx.
NEED A HINT?
Notice that 2x2x is the derivative of x2+5x^2 + 5.
SHOW DETAILED EXPLANATION

Step 1: Pick $u$

Let u=x2+5u = x^2 + 5.

Step 2: Find $du$

du=2xdxdu = 2x dx.

Step 3: Integrate

∫u3 du=14u4+C=14(x2+5)4+C\int u^3 \, du = \frac{1}{4}u^4 + C = \frac{1}{4}(x^2+5)^4 + C.
Example 02AB/BC Standard
Evaluate ∫01xex2 dx\int_0^1 x e^{x^2} \, dx.
NEED A HINT?
Don't forget to change the integration bounds when moving from xx to uu.
SHOW DETAILED EXPLANATION

Step 1: Pick $u$ and $du$

u=x2  ⟹  du=2xdxu = x^2 \implies du = 2x dx, so xdx=12dux dx = \frac{1}{2}du.

Step 2: Change Bounds

If x=0,u=02=0x=0, u=0^2=0. If x=1,u=12=1x=1, u=1^2=1.

Step 3: Solve

12∫01eu du=12[eu]01=12(e−1)\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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