Integration

CIE A-Level Mathematics· Pure Mathematics 2 & 3 (§2.5 / §3.5): Integration· 15 min read

1. Standard Integrals (Pure 2 & 3)★★☆☆☆⏱ 4 min

Across Pure 2 and Pure 3 you extend the standard integrals from Pure 1 to exponential, reciprocal and trigonometric forms of a linear argument $a x + b$ . All of these are shared by Papers 2 and 3; Pure 3 then adds one further standard result, the inverse-tangent integral $\int \frac{1}{a ^{2} + x ^{2}} d x$ .

📘 Definition

Indefinite Integral (Antiderivative)

\int f(x) dx = F(x) + C

A function $F (x)$ whose derivative equals the original integrand $f (x)$ , with $C$ the arbitrary constant of integration.

Example:

The antiderivative of $2 x$ is $x^{2} + C$

\begin{aligned} \int e^{ax+b} dx &= \frac{1}{a}e^{ax+b} + C \\ \int \frac{1}{ax+b} dx &= \frac{1}{a}\ln|ax+b| + C \\ \int \sin(ax+b) dx &= -\frac{1}{a}\cos(ax+b) + C \\ \int \cos(ax+b) dx &= \frac{1}{a}\sin(ax+b) + C \\ \int \sec^2 x dx &= \tan x + C \\ \int \frac{1}{a^2+x^2} dx &= \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C \end{aligned}

📐 Worked Example

Find the indefinite integral $\int \frac{3}{4 + x ^{2}} d x$

1
Match the integrand to the standard form for $\int \frac{1}{a ^{2} + x ^{2}} d x$ , with $a^{2} = 4$ so $a = 2$ .
2
$3 \int \frac{1}{2^2 + x^2} dx$
3
Apply the standard integral formula, adding the constant of integration:
4
$3 \times \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C = \frac{3}{2}\arctan\left(\frac{x}{2}\right) + C$

Exam tip:

Always write the constant of integration $C$ for indefinite integrals, you will lose a mark for omitting it.

2. Integration Using Trigonometric Identities★★☆☆☆⏱ 4 min

Powers of sine, cosine and tangent have no direct standard integral. Before integrating, rewrite them with a double-angle or Pythagorean identity so that only $sin (k x)$ , $cos (k x)$ or $sec^{2} x$ terms remain.

\begin{aligned} \int \sin^2 x\,dx &= \int \frac{1-\cos 2x}{2}\,dx = \frac{x}{2} - \frac{1}{4}\sin 2x + C \\ \int \cos^2 2x\,dx &= \int \frac{1+\cos 4x}{2}\,dx = \frac{x}{2} + \frac{1}{8}\sin 4x + C \\ \int \tan^2 x\,dx &= \int (\sec^2 x - 1)\,dx = \tan x - x + C \end{aligned}

📐 Worked Example

Find $\int cos^{2} 2 x d x$ .

1
There is no direct integral for $cos^{2} 2 x$ , so apply the double-angle identity $cos^{2} θ = \frac{1 + c o s 2 θ}{2}$ with $θ = 2 x$ :
2
$\cos^2 2x = \frac{1 + \cos 4x}{2}$
3
Integrate term by term, remembering to divide by the coefficient of $x$ inside the cosine:
4
$\int \frac{1+\cos 4x}{2}\,dx = \frac{1}{2}\left(x + \frac{1}{4}\sin 4x\right) + C = \frac{x}{2} + \frac{1}{8}\sin 4x + C$

Exam tip:

You cannot integrate $sin^{2} x$ , $cos^{2} 2 x$ or $tan^{2} x$ directly — always convert with a trigonometric identity first. These integrals are examinable in both Paper 2 and Paper 3.

3. Recognising kf'(x)/f(x) → ln|f(x)|★★★☆☆P3 only⏱ 4 min

When the top of a fraction is (a constant multiple of) the derivative of the bottom, the integral is a natural logarithm. This is really integration by substitution with $u = f (x)$ done by inspection, and it saves time in the exam.

\begin{aligned} \int \frac{f'(x)}{f(x)}\,dx &= \ln|f(x)| + C \\ \int \frac{kf'(x)}{f(x)}\,dx &= k\ln|f(x)| + C \\ \int \frac{2x}{x^2+3}\,dx &= \ln(x^2+3) + C \\ \int \tan x\,dx &= \int \frac{\sin x}{\cos x}\,dx = -\ln|\cos x| + C \end{aligned}

📐 Worked Example

Find $\int tan x d x$ .

1
Write $tan x$ as a quotient so the pattern is visible:
2
$\int \tan x\,dx = \int \frac{\sin x}{\cos x}\,dx$
3
The derivative of the denominator $cos x$ is $- sin x$ , so the numerator is $- 1$ times $f^{'} (x)$ with $f (x) = cos x$ :
4
$= -\int \frac{-\sin x}{\cos x}\,dx = -\ln|\cos x| + C$

Exam tip:

This is a Pure 3 technique. Before setting up a substitution, check whether the numerator is a constant multiple of the derivative of the denominator — if so, the integral is a logarithm.

4. Integration by Substitution★★★☆☆P3 only⏱ 5 min

Integration by substitution simplifies integrals of composite functions by changing the variable of integration. It is the reverse of the chain rule for differentiation.

\int f(x) dx = \int f(u(x)) \times \frac{dx}{du} du

📐 Worked Example

Use the substitution $u = x^{2} + 1$ to find $\int \frac{x}{x ^{2} + 1} d x$

1
Differentiate $u$ with respect to $x$ to find $d x$ in terms of $d u$ :
2
$u = x^2 + 1 \implies \frac{du}{dx} = 2x \implies dx = \frac{du}{2x}$
3
Substitute into the original integral, cancel common terms:
4
$\int \frac{x}{\sqrt{u}} \times \frac{du}{2x} = \int \frac{1}{2\sqrt{u}} du$
5
Integrate the simplified expression with respect to $u$ :
6
$\frac{1}{2} \int u^{-1/2} du = \frac{1}{2} \times 2 u^{1/2} + C = \sqrt{u} + C$
7
Substitute back to the original variable $x$ :
8
$= \sqrt{x^2 + 1} + C$

Exam tip:

For definite integrals, change the limits to the new variable immediately to avoid substitution errors later.

5. Integration by Parts★★★☆☆P3 only⏱ 5 min

Integration by parts is the reverse of the product rule for differentiation, used to integrate products of two different types of functions (e.g. $x sin x$ , $x e^{x}$ ).

\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx

📐 Worked Example

Find $\int x e^{2 x} d x$

1
Apply the LIATE rule: $x$ is algebraic, $e^{2 x}$ is exponential, so set $u = x$ and $\frac{d v}{d x} = e^{2 x}$ .
2
Calculate $\frac{d u}{d x}$ and $v$ :
3
$\frac{du}{dx} = 1, \quad v = \int e^{2x} dx = \frac{1}{2}e^{2x}$
4
Substitute into the integration by parts formula:
5
$uv - \int v \frac{du}{dx} dx = \frac{1}{2}x e^{2x} - \int \frac{1}{2}e^{2x} dx$
6
Integrate the remaining term and add the constant:
7
$= \frac{1}{2}x e^{2x} - \frac{1}{4}e^{2x} + C = \frac{1}{4}e^{2x}(2x - 1) + C$

Exam tip:

For repeated integration by parts, keep track of signs to avoid arithmetic errors.

6. Integration via Partial Fractions★★★★☆P3 only⏱ 6 min

To integrate a rational function (a fraction of two polynomials), you first decompose it into partial fractions, then integrate each term separately using the standard result for $\frac{1}{a x + b}$ .

📐 Worked Example

Find $\int \frac{3 x + 1}{( x - 1 ) ( x + 2 )} d x$

1
The degree of the numerator (1) is less than the degree of the denominator (2), so it is proper. Decompose into partial fractions:
2
$\frac{3x + 1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$
3
Solve for constants $A$ and $B$ by substituting roots of the denominator:
4
$3x+1 = A(x+2) + B(x-1) \implies A = \frac{4}{3}, B = \frac{5}{3}$
5
Rewrite the integral and integrate term by term:
6
$\int \left(\frac{4}{3(x-1)} + \frac{5}{3(x+2)}\right) dx$
7
$= \frac{4}{3} \ln|x-1| + \frac{5}{3} \ln|x+2| + C$

Exam tip:

Always check if the rational function is improper before decomposing into partial fractions.

7. Common Pitfalls

Wrong move:

Omitting the constant of integration $C$ for indefinite integrals

Why:

Examiners always penalize missing constants, even if the rest of the answer is correct

Correct move:

Always write $+ C$ at the end of every indefinite integral

Wrong move:

Forgetting to change limits of integration for substitution with definite integrals

Why:

This leads to common arithmetic errors when substituting back to the original variable

Correct move:

Calculate new limits for the substituted variable immediately after choosing the substitution

Wrong move:

Choosing the wrong $u$ in integration by parts, leading to a more complex integral

Why:

Picking $u$ as the exponential/trigonometric function instead of algebraic leads to unnecessary complexity

Correct move:

Always use the LIATE mnemonic to select the correct $u$

Wrong move:

Forgetting to divide by the coefficient of $x$ for composite functions, e.g. $\int e^{2 x} d x = e^{2 x} + C$

Why:

You forget to apply the reverse chain rule for linear inner functions

Correct move:

Always divide by the derivative of the linear term, so $\int e^{2 x} d x = \frac{1}{2} e^{2 x} + C$

Wrong move:

Skipping polynomial division for improper rational functions

Why:

You cannot decompose an improper rational function into partial fractions correctly, leading to an incorrect result

Correct move:

Always check the degrees of numerator and denominator, divide first if the fraction is improper

8. Quick Reference Cheatsheet

Integral Type	Formula / Rule	Key Notes
Composite $f (a x + b)$	$\frac{1}{a} F (a x + b) + C$	Divide by $a$ , add $C$
Integration by substitution	$\int f (x) d x = \int f (u (x)) \frac{d x}{d u} d u$	Change limits for definite integrals
Integration by parts	$\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x$	Choose $u$ by LIATE rule
$\int \frac{1}{a x + b} d x$	$\frac{1}{a} ln ∣ a x + b ∣ + C$	Keep absolute value
$\int \frac{1}{a ^{2} + x ^{2}} d x$	$\frac{1}{a} arctan (\frac{x}{a}) + C$	Recall for P3
Improper rational function	Divide first, then decompose to partial fractions	Don't skip the division step
$\int sin^{2} x d x$	$\frac{x}{2} - \frac{1}{4} sin 2 x + C$	Use $sin^{2} x = \frac{1 - c o s 2 x}{2}$
$\int cos^{2} 2 x d x$	$\frac{x}{2} + \frac{1}{8} sin 4 x + C$	Use $cos^{2} 2 x = \frac{1 + c o s 4 x}{2}$
$\int tan^{2} x d x$	$tan x - x + C$	Use $tan^{2} x = sec^{2} x - 1$
$\int \frac{k f ^{'} ( x )}{f ( x )} d x$	$k ln ∣ f (x) ∣ + C$	Top = derivative of bottom (P3)

9. Frequently Asked

Do I need to memorise all the standard integral results in Pure 2 and 3?

Yes, most standard integrals in Pure 2 and 3 are not given in the formula booklet, so you are expected to recall them.

Do I have to show full working for substitution steps?

Yes, full marks require you to show all substitution steps, even if you arrive at the correct final answer.

What's Next

Mastering these integration techniques underpins almost all remaining calculus in CIE Pure 2 & 3, including solving differential equations and further applications. You will also combine them with the volume-of-revolution method you already met in Pure 1 (§1.8) — for example integrating a partial-fraction or trigonometric-identity expression to find a volume. Integration is weighted heavily in Papers 2 and 3, and questions frequently combine several techniques (e.g. substitution after partial fractions, or repeated integration by parts) in one problem. Building fluency now will help you tackle harder, higher-mark questions efficiently.