Integration by substitution

IB Mathematics AI HL· 20 min read

1. u-Substitution for Indefinite and Definite Integrals★★☆☆☆⏱ 8 min

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📘 Definition

u-Substitution

A technique that reverses the chain rule, used to integrate composite functions by changing the variable of integration to simplify the integrand.

Example:

$\int 2 x cos (x^{2}) d x$

To apply u-substitution, select $u$ as the inner function of a composite term. The derivative of $u$ should cancel a term from the original integrand. For definite integrals, you must adjust the bounds of integration to match the new variable $u$ .

📐 Worked Example

Evaluate $\int_{1}^{2} 2 x e^{x^{2}} d x$

1
Choose the inner function as $u$ :
2
$u = x^2$
3
Differentiate $u$ to find $d u$ :
4
$\frac{du}{dx} = 2x \implies du = 2x dx$
5
Adjust limits of integration for the new variable $u$ :
6
When $x = 1$ , $u = 1^{2} = 1$ ; when $x = 2$ , $u = 2^{2} = 4$
7
Substitute into the original integral:
8
$\int_{1}^{4} e^u du$
9
Integrate and evaluate:
10
$[e^u]_1^4 = e^4 - e = e(e^3 - 1) \approx 51.93$

Exam tip:

Always check your antiderivative by differentiating it to confirm it matches the original integrand.

2. Common Pitfalls

Wrong move:

Forgetting to adjust limits of integration for u-substitution on definite integrals

Why:

You will end up evaluating the wrong expression with the original bounds, leading to an incorrect result

Correct move:

Calculate new upper and lower bounds for $u$ immediately after defining $u$ , and evaluate the integral entirely in terms of $u$

Wrong move:

Omitting the constant of integration $C$ for indefinite integrals

Why:

Antiderivatives are families of functions, and omitting $C$ will lose marks in exams

Correct move:

Always add $+ C$ to the final result for any indefinite integral

3. Quick Reference Cheatsheet

Technique	Key Rule/Formula	When to Use
u-Substitution	$\int f (g (x)) g^{'} (x) d x = \int f (u) d u$ , $u = g (x)$	Composite function, function × derivative
Definite u-Substitution	Change bounds for $u$ , evaluate directly	Avoid substituting back to $x$

What's Next

Mastering u-substitution is foundational for the applications of integration in IB AI HL. These skills are required for calculating areas between curves, volumes of revolution, and solving separable differential equations, all of which make up a large portion of exam marks. Regular practice recognising when an integrand can be simplified by a substitution is the best way to prepare for these exam questions.