Integration (Edexcel IAL Math P3 WMA13)

Edexcel International A-Level Mathematics· 2018 Specification Issue 3, Unit P3 §5· 20 min read

1. Standard Integrals for P3★★☆☆☆⏱ 5 min

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.

📘 Definition

Standard Integrals

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

Example:

$\int 3 cos 2 x d x = \frac{3}{2} sin 2 x + c$

$\int e^{k x} d x = \frac{1}{k} e^{k x} + c$
$\int \frac{1}{x} d x = ln ∣ x ∣ + c, x \neq = 0$
$\int sin k x d x = - \frac{1}{k} cos k x + c$
$\int cos k x d x = \frac{1}{k} sin k x + c$
$\int a^{x} d x = \frac{a ^{x}}{l n a} + c$

📐 Worked Example

Find $\int (2 e^{5 x} + \frac{3}{2 x} - 4 sin 3 x) d x$

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 $1/ (k x)$ , factor out $1/ k$ first to avoid mistakes: $\int 1/ (2 x) d x = (1/2) ln ∣ x ∣ + c$ , not $2 ln ∣ x ∣ + c$ .

2. Integration by Recognition: Reverse Chain Rule★★★☆☆⏱ 6 min

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.

📘 Definition

Recognition Integral Forms

$\int f'(x)[f(x)]^n dx = \frac{[f(x)]^{n+1}}{n+1} + c, n \neq -1$,$\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + c$

Two standard reverse chain rule structures that can be integrated without substitution.

📐 Worked Example

Find $\int 10 x (3 x^{2} + 7)^{4} d x$

1
Identify $f (x) = 3 x^{2} + 7$ , so $f^{'} (x) = 6 x$ .
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 $\int \frac{6 c o s 2 x}{s i n 2 x + 5} d x$

1
Identify $f (x) = sin 2 x + 5$ , so $f^{'} (x) = 2 cos 2 x$ .
2
The numerator is $6 cos 2 x = 3 \times 2 cos 2 x = 3 f^{'} (x)$ . 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.

3. Integrating Trigonometric Functions Using Identities★★★★☆⏱ 5 min

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.

📘 Definition

Key Trig Identities for P3 Integration

Double angle identities and Pythagorean identities used to rewrite squared trigonometric functions into integrable linear forms.

Example:

$sin^{2} x = \frac{1 - c o s 2 x}{2}$ , $cos^{2} x = \frac{1 + c o s 2 x}{2}$ , $tan^{2} x = sec^{2} x - 1$

📐 Worked Example

Find $\int tan^{2} 3 x d x$

1
Use the Pythagorean identity $tan^{2} θ = sec^{2} θ - 1$ , with $θ = 3 x$ :
2
$\int (\sec^2 3x - 1) dx$
3
Integrate each term separately. Use the formula booklet result for $\int sec^{2} k x d x$ :
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 $\int 4 cos^{2} 2 x d x$

1
Apply the double angle identity for cosine squared: $cos^{2} θ = \frac{1 + c o s 2 θ}{2}$ , with $θ = 2 x$ :
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.

4. Exam-Style Mixed Integration Practice★★★★☆⏱ 4 min

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 $\int (tan x + 3 cos^{2} x) d x$ 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 $\int tan x d x = ln ∣ sec x ∣ + c_{1}$
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
$\ln|\sec x| + \frac{3}{2}x + \frac{3}{4}\sin 2x + c$

✓ Quick check

What is the value of $\int 2 sec^{2} 5 x d x$ ?
Reveal answer
$\frac{2}{5}\tan 5x + c$ —
Correct! This uses the formula booklet result for $\int sec^{2} k x d x = (1/ k) tan k x + c$ .
What identity would you use to integrate $sin^{2} 4 x$ ?
Reveal answer
Double angle identity for cosine: $\sin^2 \theta = (1 - \cos 2\theta)/2$ —
Correct. This rewrites the squared sine term into a linear form you can integrate directly.

5. Common Pitfalls

Wrong move:

Forgetting the constant of integration $c$ at the end of indefinite integrals

Why:

Edexcel deducts 1 mark per missing constant in P3 integration questions

Correct move:

Add $+ c$ as the final step for every indefinite integral, regardless of the method used

Wrong move:

Omitting absolute value signs around the argument of $ln$ when integrating $1/ x$ or $f^{'} (x) / f (x)$

Why:

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 move:

Always write $ln ∣ f (x) ∣$ for these integral forms, not $ln (f (x))$

Wrong move:

Using the wrong sign when integrating $sin k x$

Why:

The derivative of $cos k x$ is $- k sin k x$ , so the integral of $sin k x$ requires a negative sign

Correct move:

Memorise $\int sin k x d x = - \frac{1}{k} cos k x + c$ , and double check by differentiating your result

Wrong move:

Incorrectly integrating $1/ (k x)$ as $k ln ∣ x ∣ + c$

Why:

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 move:

Rewrite $1/ (k x) = (1/ k) (1/ x)$ , so the integral is $(1/ k) ln ∣ x ∣ + c$

Wrong move:

Attempting to use integration by substitution or parts for P3 recognition integrals

Why:

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 move:

Use the two recognition integral forms for reverse chain rule problems in P3

6. Quick Reference Cheatsheet

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

7. Frequently Asked

Do I need to write absolute value signs for ln integrals?

Yes, you must include absolute value signs around the argument of the natural logarithm when integrating $\frac{1}{x}$ or $\frac{f ^{'} ( x )}{f ( x )}$ , as the logarithm is only defined for positive values. You will lose 1 mark per omission in Edexcel exams.

Can I use integration by substitution for P3 recognition integrals?

No, you are expected to use integration by recognition for these problems in P3. Substitution is a P4 topic, and using it when recognition is expected will waste time and may not earn full marks if working is incomplete.

Going deeper

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.