Integration by Substitution (u-Substitution)

By the end of this lesson, you will be able to apply $u$-substitution to evaluate indefinite and definite integrals by reversing the Chain Rule.

The Big Idea

The Chain Rule says $(F(g(x)))' = F'(g(x)) \cdot g'(x)$. In integration, this reverses as:

$$\int F'(g(x)) g'(x),dx = F(g(x)) + C$$

The $u$-substitution: let $u = g(x)$, then $du = g'(x),dx$.

The Substitution Process

1. Choose $u$: identify an inner function $u = g(x)$
2. Compute $du$: $du = g'(x),dx$
3. Substitute: replace $g(x)$ with $u$ and $g'(x),dx$ with $du$
4. Integrate in terms of $u$
5. Back-substitute: replace $u$ with $g(x)$

Example 1 — Basic Substitution

$$\int 2x(x^2+1)^5,dx$$

Let $u = x^2+1$, $du = 2x,dx$:

$$= \int u^5,du = \frac{u^6}{6} + C = \frac{(x^2+1)^6}{6} + C$$

Example 2 — Finding the Right $u$

$$\int \frac{x}{\sqrt{1-x^2}},dx$$

Let $u = 1-x^2$, $du = -2x,dx \implies x,dx = -du/2$:

$$= \int\frac{-du/2}{\sqrt{u}} = -\frac{1}{2}\int u^{-1/2},du = -\frac{1}{2}(2\sqrt{u}) + C = -\sqrt{1-x^2} + C$$

Example 3 — Trig Substitution

$$\int\sin^3 x\cos x,dx$$

Let $u = \sin x$, $du = \cos x,dx$:

$$= \int u^3,du = \frac{u^4}{4} + C = \frac{\sin^4 x}{4} + C$$

Example 4 — Exponential

$$\int xe^{x^2},dx$$

Let $u = x^2$, $du = 2x,dx \implies x,dx = du/2$:

$$= \frac{1}{2}\int e^u,du = \frac{1}{2}e^{x^2} + C$$

Definite Integrals with Substitution

Method 1: substitute back to $x$ form and evaluate at original limits.

Method 2 (preferred): change the limits to $u$-values.

Example 5 — Definite Integral with Limit Change

$$\int_0^2 x(x^2+1)^3,dx$$

$u = x^2+1$; when $x=0$: $u=1$; when $x=2$: $u=5$.

$du = 2x,dx \implies x,dx = du/2$:

$$= \frac{1}{2}\int_1^5 u^3,du = \frac{1}{2}\left[\frac{u^4}{4}\right]_1^5 = \frac{1}{8}[625-1] = \frac{624}{8} = 78$$

Example 6 — Recognizing the Pattern

$$\int\frac{\ln x}{x},dx$$

$u = \ln x$, $du = dx/x$: