Introduction to Integration by Substitution

Integration by substitution (sometimes called u-substitution) is the integration counterpart of the chain rule. It transforms a complicated integral into a simpler one by replacing a portion of the integrand with a new variable.

The core idea: if your integral has the structure

$\int k f (g (x)) g^{'} (x) d x$

you can let $u = g (x)$ , so that $d u = g^{'} (x) d x$ . The integral then becomes:

$k \int f (u) d u$

which — if you chose $u$ wisely — is much easier to evaluate.

Integration by Substitution

A technique that simplifies an integral by replacing a composite expression $g (x)$ with a new variable $u$ , converting $\int f (g (x)) g^{'} (x) d x$ into $\int f (u) d u$ .

Note

In IB examinations, if the integral is not already in the form $\int k g^{'} (x) f (g (x)) d x$ , the substitution will be provided for you in the question. When it is provided, your job is to apply it correctly — not to invent it.

The method only helps if the resulting integral in $u$ is simpler. Always think before choosing your substitution: will this actually make things easier?

Steps for Integration by Substitution

Follow these five steps systematically:

Identify a suitable substitution $u = g (x)$ — look for an inner function whose derivative also appears in the integrand.
Differentiate to find $\frac{d u}{d x} = g^{'} (x)$ , then rearrange to express $d x$ (or $g^{'} (x) d x$ ) in terms of $d u$ .
Rewrite the entire integral in terms of $u$ only — no $x$ should remain.
Integrate with respect to $u$ .
Back-substitute to express the answer in terms of $x$ .

Exam Tip

Look for a function and its derivative both present in the integrand — that pairing is almost always the right choice for $u$ . For example, in $\int x e^{x^{2}} d x$ , notice that $x$ is (up to a constant) the derivative of $x^{2}$ .

Warning

For definite integrals, when you substitute $u = g (x)$ you must also change the limits. The lower limit $x = a$ becomes $u = g (a)$ and the upper limit $x = b$ becomes $u = g (b)$ . Failing to change the limits is one of the most common errors in exam scripts.

Example

Example: Evaluate $\int x 1 - x^{2} d x$

Step 1: Let $u = 1 - x^{2}$

Step 2: $\frac{d u}{d x} = - 2 x ⟹ d u = - 2 x d x ⟹ x d x = - \frac{1}{2} d u$

Step 3: The integral becomes:
$- \frac{1}{2} \int u d u = - \frac{1}{2} \int u^{1/2} d u$

Step 4: Integrate:
$- \frac{1}{2} \cdot \frac{2}{3} u^{3/2} + C = - \frac{1}{3} u^{3/2} + C$

Step 5: Back-substitute $u = 1 - x^{2}$ :
$- \frac{1}{3} (1 - x^{2})^{3/2} + C$

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AHL 5.16—Integration by substitution, parts and repeated parts

Introduction to Integration by Substitution

Steps for Integration by Substitution

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