# Integration (Edexcel IAL Math P3 WMA13)

> Edexcel International A-Level Mathematics · P3 (WMA13)
> Source: https://www.owlsprep.com/study/edexcel-ial-math-p3-integration/

This guide covers all integration content required for Edexcel IAL Pure Maths 3 (WMA13), including standard integral rules, integration by recognition, and trigonometric integration using identities, tailored for IA2 exam preparation.

**Prerequisites:** [Differentiation of standard functions (P2 topic)](https://www.owlsprep.com/study/edexcel-ial-math-p2-differentiation/); [Trigonometric identities (P2 topic)](https://www.owlsprep.com/study/edexcel-ial-math-p2-trigonometric-identities/)

## Learning objectives

- Integrate standard functions including $e^{kx}$, $\frac{1}{x}$, $\sin kx$ and $\cos kx$, and their sums and differences
- Apply integration by recognition for $\int f'(x)/f(x) dx$ and $\int f'(x)[f(x)]^n dx$ forms
- Use trigonometric identities to rewrite squared trig terms for integration
- Correctly include absolute value signs and constant of integration for all indefinite integrals

## Standard Integrals for P3

The core set of standard integrals you are required to know for P3 are listed below. All are indefinite integrals, so you must add a constant of integration $c$ to every final result.

**Standard Integrals** — A set of antiderivatives of common functions that can be applied directly, or to linear combinations of functions.

*Example:* $\int 3\cos 2x dx = \frac{3}{2}\sin 2x + c$

- $\int e^{kx} dx = \frac{1}{k}e^{kx} + c$
- $\int \frac{1}{x} dx = \ln|x| + c, x \neq 0$
- $\int \sin kx dx = -\frac{1}{k}\cos kx + c$
- $\int \cos kx dx = \frac{1}{k}\sin kx + c$
- $\int a^x dx = \frac{a^x}{\ln a} + c$

> **note**
>
> The following integrals are provided in your exam formula booklet, so you do not need to memorise them: $\int \sec^2 kx dx = \frac{1}{k}\tan kx + c$, $\int \tan x dx = \ln|\sec x| + c$, $\int \cot x dx = \ln|\sin x| + c$.

**Worked example:** Find $\int \left( 2e^{5x} + \frac{3}{2x} - 4\sin 3x \right) dx$

1. Integrate each term separately using the standard integral rules:
2. $$\int 2e^{5x} dx = 2 \times \frac{1}{5}e^{5x} = \frac{2}{5}e^{5x}$$
3. $$\int \frac{3}{2x} dx = \frac{3}{2} \int \frac{1}{x} dx = \frac{3}{2}\ln|x|$$
4. $$\int -4\sin 3x dx = -4 \times \left(-\frac{1}{3}\cos 3x\right) = \frac{4}{3}\cos 3x$$
5. Combine all results and add the constant of integration $c$:
6. $$\frac{2}{5}e^{5x} + \frac{3}{2}\ln|x| + \frac{4}{3}\cos 3x + c$$

> **Exam tip:** When integrating $1/(kx)$, factor out $1/k$ first to avoid mistakes: $\int 1/(2x) dx = (1/2)\ln|x| + c$, not $2\ln|x| + c$.

## Integration by Recognition: Reverse Chain Rule

Integration by recognition, also called the reverse chain rule, lets you integrate functions that are structured as the derivative of a composite function. There are two key forms you must master for P3.

**Recognition Integral Forms** — Two standard reverse chain rule structures that can be integrated without substitution.

> **tip**
>
> If the multiple of $f'(x)$ in your integrand is not 1, you can adjust by multiplying by the reciprocal of the constant to make the integral match the required form.

**Worked example:** Find $\int 10x(3x^2 + 7)^4 dx$

1. Identify $f(x) = 3x^2 + 7$, so $f'(x) = 6x$.
2. We have 10x in the integral, which is $\frac{10}{6} \times f'(x) = \frac{5}{3}f'(x)$. Rewrite the integral:
3. $$\int \frac{5}{3} \times 6x(3x^2 +7)^4 dx = \frac{5}{3} \int f'(x)[f(x)]^4 dx$$
4. Apply the first recognition rule with $n=4$:
5. $$\frac{5}{3} \times \frac{[f(x)]^5}{5} + c = \frac{1}{3}(3x^2 +7)^5 + c$$

**Worked example:** Find $\int \frac{6\cos 2x}{\sin 2x + 5} dx$

1. Identify $f(x) = \sin 2x + 5$, so $f'(x) = 2\cos 2x$.
2. The numerator is $6\cos 2x = 3 \times 2\cos 2x = 3f'(x)$. Rewrite the integral:
3. $$3 \int \frac{f'(x)}{f(x)} dx$$
4. Apply the second recognition rule:
5. $$3\ln|f(x)| + c = 3\ln|\sin 2x + 5| + c$$

> **Exam tip:** Check your recognition integral result by differentiating it: if you get back the original integrand, your answer is correct. This is a fast way to catch errors in the exam.

## Integrating Trigonometric Functions Using Identities

Many trigonometric integrals cannot be solved directly with standard rules, so you first need to rewrite the integrand using a trigonometric identity to convert it into a form you can integrate.

**Key Trig Identities for P3 Integration** — Double angle identities and Pythagorean identities used to rewrite squared trigonometric functions into integrable linear forms.

*Example:* $\sin^2 x = \frac{1 - \cos 2x}{2}$, $\cos^2 x = \frac{1 + \cos 2x}{2}$, $\tan^2 x = \sec^2 x - 1$

**Worked example:** Find $\int \tan^2 3x dx$

1. Use the Pythagorean identity $\tan^2 \theta = \sec^2 \theta - 1$, with $\theta = 3x$:
2. $$\int (\sec^2 3x - 1) dx$$
3. Integrate each term separately. Use the formula booklet result for $\int \sec^2 kx dx$:
4. $$\int \sec^2 3x dx = \frac{1}{3}\tan 3x$$
5. $$\int -1 dx = -x$$
6. Combine results and add constant of integration:
7. $$\frac{1}{3}\tan 3x - x + c$$

**Worked example:** Find $\int 4\cos^2 2x dx$

1. Apply the double angle identity for cosine squared: $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$, with $\theta = 2x$:
2. $$4 \times \int \frac{1 + \cos 4x}{2} dx = 2 \int (1 + \cos 4x) dx$$
3. Integrate term by term:
4. $$2 \left( x + \frac{1}{4}\sin 4x \right) + c$$
5. Simplify the final result:
6. $$2x + \frac{1}{2}\sin 4x + c$$

> **Exam tip:** Always write down the identity you are using first to show your working. Edexcel awards marks for correct identity application even if you make an arithmetic error later.

## Exam-Style Mixed Integration Practice

P3 exam questions will often combine multiple integration techniques in one problem, requiring you to select the correct rule or identity for each part of the integrand.

**Worked example:** Simplify $\int \left( \tan x + 3\cos^2 x \right) dx$ to a form ready for integration (you do not need to evaluate P4 content)

1. Split the integral into two separate terms:
2. $$\int \tan x dx + \int 3\cos^2 x dx$$
3. First term: Use the formula booklet result for $\int \tan x dx = \ln|\sec x| + c_1$
4. Second term: Rewrite $\cos^2 x$ using the double angle identity:
5. $$3 \times \int \frac{1 + \cos 2x}{2} dx = \frac{3}{2} \int 1 dx + \frac{3}{2} \int \cos 2x dx$$
6. Integrate the standard terms:
7. $$\frac{3}{2}x + \frac{3}{4}\sin 2x + c_2$$
8. Combine all results, merging constants into a single $c$:
9. $$\ln|\sec x| + \frac{3}{2}x + \frac{3}{4}\sin 2x + c$$

**Check your understanding**

1. What is the value of $\int 2\sec^2 5x dx$?

   *Why:* Correct! This uses the formula booklet result for $\int \sec^2 kx dx = (1/k)\tan kx + c$.

2. What identity would you use to integrate $\sin^2 4x$?

   *Why:* Correct. This rewrites the squared sine term into a linear form you can integrate directly.

## Common pitfalls

- **Wrong:** Forgetting the constant of integration $c$ at the end of indefinite integrals
  - Why it fails: Edexcel deducts 1 mark per missing constant in P3 integration questions
  - Correct: Add $+ c$ as the final step for every indefinite integral, regardless of the method used
- **Wrong:** Omitting absolute value signs around the argument of $\ln$ when integrating $1/x$ or $f'(x)/f(x)$
  - Why it fails: The natural logarithm is only defined for positive values, so absolute values are required for mathematical correctness, and marks are deducted for missing them
  - Correct: Always write $\ln|f(x)|$ for these integral forms, not $\ln(f(x))$
- **Wrong:** Using the wrong sign when integrating $\sin kx$
  - Why it fails: The derivative of $\cos kx$ is $-k\sin kx$, so the integral of $\sin kx$ requires a negative sign
  - Correct: Memorise $\int \sin kx dx = -\frac{1}{k}\cos kx + c$, and double check by differentiating your result
- **Wrong:** Incorrectly integrating $1/(kx)$ as $k\ln|x| + c$
  - Why it fails: The coefficient of x in the denominator is a constant factor that can be taken outside the integral, so it ends up in the numerator of the fraction
  - Correct: Rewrite $1/(kx) = (1/k)(1/x)$, so the integral is $(1/k)\ln|x| + c$
- **Wrong:** Attempting to use integration by substitution or parts for P3 recognition integrals
  - Why it fails: These methods are part of P4 content, and using them wastes time in the exam, and may not earn full marks if working is incomplete
  - Correct: Use the two recognition integral forms for reverse chain rule problems in P3

## Cheatsheet

| Integral Type | Rule | Notes |
| --- | --- | --- |
| $\int e^{kx} dx$ | $\frac{1}{k}e^{kx} + c$ | Memorise |
| $\int \frac{1}{x} dx$ | $\ln\|x\| + c$ | Include absolute value |
| $\int \sin kx dx$ | $-\frac{1}{k}\cos kx + c$ | Negative sign required |
| $\int \cos kx dx$ | $\frac{1}{k}\sin kx + c$ | No negative sign |
| $\int f'(x)[f(x)]^n dx$ | $\frac{[f(x)]^{n+1}}{n+1} + c, n \neq -1$ | Recognition (reverse chain) |
| $\int \frac{f'(x)}{f(x)} dx$ | $\ln\|f(x)\| + c$ | Absolute value required |
| $\int \tan^2 kx dx$ | $\frac{1}{k}\tan kx - x + c$ | Use $\tan^2 = \sec^2 - 1$ |
| $\int \sin^2 kx dx$ | $\frac{x}{2} - \frac{\sin 2kx}{4k} + c$ | Use double angle identity |

## What's next

Now that you have mastered P3 integration content, you are ready to progress to more advanced integration techniques in Pure Maths 4 (P4), including integration by substitution, by parts, and using partial fractions. These skills build directly on the recognition integral rules you learned in P3, so ensure you are confident identifying $f(x)$ and $f'(x)$ in composite functions before moving on. You will also use these integration skills to solve high-mark exam problems involving volumes of revolution and separable differential equations in P4. For additional P3 practice, attempt past paper integration questions from WMA13 papers, focusing only on questions requiring standard rules, recognition integrals and trig identities, as definite integrals and area problems are P2 content.

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