# Integration by substitution

> IB Mathematics AI HL · Calculus
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u4-integration-techniques-substitution-and-by/

This module covers u-substitution, a core integration technique for IB AI HL used to integrate composite functions. You will learn to apply it to both indefinite and definite integrals, including how to adjust the limits of integration for definite integrals.

**Prerequisites:** [Basic antiderivatives of common functions](https://www.owlsprep.com/study/ib-math-ai-hl-u3-basic-antiderivatives/); [Chain rule for differentiation](https://www.owlsprep.com/study/ib-math-ai-hl-u3-differentiation-rules/)

## Learning objectives

- Recognise when u-substitution is the appropriate integration technique
- Apply u-substitution to evaluate indefinite integrals of composite functions
- Adjust the limits of integration when using u-substitution for definite integrals
- Avoid common errors in variable selection and limit adjustment

## u-Substitution for Indefinite and Definite Integrals

**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:* $\int 2x \cos(x^2) dx$

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 $\int_1^2 2x e^{x^2} dx$

1. Choose the inner function as $u$:
2. $$u = x^2$$
3. Differentiate $u$ to find $du$:
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.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to adjust limits of integration for u-substitution on definite integrals
  - Why it fails: You will end up evaluating the wrong expression with the original bounds, leading to an incorrect result
  - Correct: Calculate new upper and lower bounds for $u$ immediately after defining $u$, and evaluate the integral entirely in terms of $u$
- **Wrong:** Omitting the constant of integration $C$ for indefinite integrals
  - Why it fails: Antiderivatives are families of functions, and omitting $C$ will lose marks in exams
  - Correct: Always add $+C$ to the final result for any indefinite integral

## Cheatsheet

| Technique | Key Rule/Formula | When to Use |
| --- | --- | --- |
| u-Substitution | $\int f(g(x))g'(x)dx = \int f(u)du$, $u=g(x)$ | 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.

- [Area between curves, volumes of revolution](https://www.owlsprep.com/study/ib-math-ai-hl-u4-area-between-curves-volumes-of/)
- [First order differential equations: Euler's method](https://www.owlsprep.com/study/ib-math-ai-hl-u4-first-order-differential-equations-euler/)
- [Separation of variables for differential equations](https://www.owlsprep.com/study/ib-math-ai-hl-u4-separation-of-variables-for-differential/)

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