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# A-Level Maths | Integration
- URL: https://www.a-level-maths-tutor.com/a-level-maths-integration/
- Published: 2026-04-23T10:56:28.000Z
- Updated: 2026-04-23T11:46:16.000Z
- Author: Jaisul Naik
- Tags: Pure Maths

## Key Formulae

$\\displaystyle\\int e^{kx}\\,dx = \\frac{1}{k}e^{kx} + c$

$\\displaystyle\\int \\sin(kx)\\,dx = -\\frac{1}{k}\\cos(kx) + c$

$\\displaystyle\\int \\cos(kx)\\,dx = \\frac{1}{k}\\sin(kx) + c$

$\\displaystyle\\int \\frac{1}{kx+c}\\,dx = \\frac{1}{k}\\ln|kx+c| + C$

## Question 1 — Integrating Trig Functions

(a) $\\displaystyle\\int 5\\sin 2x\\,dx$

(b) $\\displaystyle\\int 3\\cos 6x\\,dx$

(c) $\\displaystyle\\int 5\\sin x - 4\\cos 2x\\,dx$

(d) $\\displaystyle\\int 5\\cos 2x - 3\\sin 5x\\,dx$

(e) $\\displaystyle\\int 15\\cos 3x - 15\\sin 5x\\,dx$

(f) $\\displaystyle\\int \\sin 8x - \\tfrac{1}{2}\\cos 3x\\,dx$

(g) $\\displaystyle\\int 2\\cos\\tfrac{1}{3}x + 3\\sin\\tfrac{1}{2}x\\,dx$

(h) $\\displaystyle\\int 7\\cos 3x - 3\\cos 7x\\,dx$

(i) $\\displaystyle\\int \\tfrac{1}{2}\\sin 5x - \\tfrac{1}{2}\\sin\\tfrac{1}{4}x\\,dx$

(j) $\\displaystyle\\int 10\\cos 2x - \\sin\\tfrac{x}{4} + 9\\sin\\tfrac{3x}{2}\\,dx$

## Question 2 — Integrating Exponential Functions

(a) $\\displaystyle\\int e^x + e^{2x} + e^{-x}\\,dx$

(b) $\\displaystyle\\int 4e^{2x} - e^{-2x} + 3e^{3x}\\,dx$

(c) $\\displaystyle\\int 2e^{4x} - e^{-3x} + \\tfrac{1}{2}e^{2x}\\,dx$

(d) $\\displaystyle\\int 4e^{-2x} - 2e^{-4x} + \\tfrac{1}{2}e^{3x}\\,dx$

(e) $\\displaystyle\\int 5e^{\\frac{1}{2}x} - \\tfrac{1}{2}e^{-\\frac{1}{2}x} + \\tfrac{3}{4}e^{\\frac{1}{4}x}\\,dx$

(f) $\\displaystyle\\int e^x + e^{2x} + e^{3x}\\,dx$

(g) $\\displaystyle\\int 6e^{2x} + e^{-2x} - 3e^{-x}\\,dx$

(h) $\\displaystyle\\int 3e^{2x} - 2e^{-2x} + \\tfrac{1}{2}e^{4x}\\,dx$

(i) $\\displaystyle\\int 6e^{-3x} - 2e^{-2x} + \\tfrac{1}{3}e^{2x}\\,dx$

(j) $\\displaystyle\\int 3e^{\\frac{1}{2}x} - \\tfrac{1}{2}e^{-\\frac{1}{4}x} + 3e^{\\frac{3}{2}x}\\,dx$

## Question 3 — Integrating Logarithmic Forms

(a) $\\displaystyle\\int \\frac{1}{x+1} + \\frac{1}{2x-1} + \\frac{1}{2-x}\\,dx$

(b) $\\displaystyle\\int \\frac{4}{2x+1} + \\frac{2}{2x-1} + \\frac{1}{1-3x}\\,dx$

(c) $\\displaystyle\\int \\frac{6}{2x-1} + \\frac{4}{3x-1} - \\frac{2}{1-4x} + \\frac{1}{2x}\\,dx$

(d) $\\displaystyle\\int \\frac{2}{3x-2} + \\frac{2}{5x-1} - \\frac{2}{1-x} + \\frac{4}{3x}\\,dx$

(e) $\\displaystyle\\int \\frac{4}{2x-3} - \\frac{2}{1-2x} - \\frac{1}{1+2x} + \\frac{1}{2x}\\,dx$

(f) $\\displaystyle\\int \\frac{1}{x+2} + \\frac{1}{3x-1} + \\frac{1}{1-x}\\,dx$

(g) $\\displaystyle\\int \\frac{6}{3x+1} + \\frac{4}{2x-1} + \\frac{1}{1-4x}\\,dx$

(h) $\\displaystyle\\int \\frac{8}{4x-1} + \\frac{5}{2x-1} - \\frac{2}{1-3x} + \\frac{4}{x}\\,dx$

(i) $\\displaystyle\\int \\frac{9}{3x-1} + \\frac{2}{6x-1} - \\frac{2}{1-2x} + \\frac{1}{2x}\\,dx$

(j) $\\displaystyle\\int \\frac{3}{5x-3} - \\frac{2}{x} - \\frac{1}{1+3x} + \\frac{9}{2x}\\,dx$

## Question 4 — Reverse Chain Rule

Integrate with respect to $x$:

(a) $(x-2)^7$

(b) $(2x+5)^3$

(c) $6(1+3x)^4$

(d) $\\left(\\tfrac{1}{4}x-2\\right)^5$

(e) $(8-5x)^4$

(f) $\\dfrac{1}{(x+7)^2}$

(g) $\\dfrac{8}{(4x-3)^5}$

(h) $\\dfrac{1}{2(5-3x)^3}$

## Question 5 — Integration by Substitution (1 step)

Showing your working in full, use the given substitution to find:

(a) $\\displaystyle\\int 2x(x^2-1)^3\\,dx$ $u = x^2-1$

(b) $\\displaystyle\\int \\sin^4 x\\cos x\\,dx$ $u = \\sin x$

(c) $\\displaystyle\\int 3x^2(2+x^3)^2\\,dx$ $u = 2+x^3$

(d) $\\displaystyle\\int 2xe^{x^2}\\,dx$ $u = x^2$

(e) $\\displaystyle\\int \\frac{x}{(x^2+3)^4}\\,dx$ $u = x^2+3$

(f) $\\displaystyle\\int \\sin 2x\\cos^3 2x\\,dx$ $u = \\cos 2x$

(g) $\\displaystyle\\int \\frac{3x}{x^2-2}\\,dx$ $u = x^2-2$

(h) $\\displaystyle\\int x\\sqrt{1-x^2}\\,dx$ $u = 1-x^2$

(i) $\\displaystyle\\int \\sec^3 x\\tan x\\,dx$ $u = \\sec x$

(j) $\\displaystyle\\int (x+1)(x^2+2x)^3\\,dx$ $u = x^2+2x$

## Question 6 — Integration by Substitution (2 steps)

Using the given substitution, find:

(a) $\\displaystyle\\int x(2x-1)^4\\,dx$ $u = 2x-1$

(b) $\\displaystyle\\int x\\sqrt{1-x}\\,dx$ $u^2 = 1-x$

(c) $\\displaystyle\\int \\frac{1}{(1-x^2)^{3/2}}\\,dx$ $x = \\sin u$

(d) $\\displaystyle\\int \\frac{1}{\\sqrt{x}-1}\\,dx$ $x = u^2$

(e) $\\displaystyle\\int (x+1)(2x+3)^3\\,dx$ $u = 2x+3$

(f) $\\displaystyle\\int \\frac{x^2}{\\sqrt{x-2}}\\,dx$ $u^2 = x-2$

## Question 7 — Integration by Substitution (with Limits)

Using the given substitution, evaluate:

(a) $\\displaystyle\\int\_1^2 x(x^2-3)^3\\,dx$ $u = x^2-3$

(b) $\\displaystyle\\int\_0^{\\pi/6} \\sin^3 x\\cos x\\,dx$ $u = \\sin x$

## Question 8 — Integrating with Partial Fractions

1\. $\\displaystyle\\int \\frac{17-4x}{(x-2)(x+1)}\\,dx$

2\. $\\displaystyle\\int \\frac{2-x}{(x+1)(2x-1)}\\,dx$

3\. $\\displaystyle\\int \\frac{4}{(x-2)(2-3x)}\\,dx$

4\. $\\displaystyle\\int \\frac{5x-7}{(x-1)(5x-3)}\\,dx$

5\. $\\displaystyle\\int \\frac{18x-1}{(2x+1)(3x-1)}\\,dx$

## Question 9 — Integration by Parts

$$\\int u\\,v' = uv - \\int v\\,u'$$

Use the **LATE** rule: Logarithmic → Algebraic → Trigonometric → Exponential. Whatever comes first is $u$; whatever comes last is $v'$.

Use integration by parts to find:

(a) $\\displaystyle\\int xe^x\\,dx$

(b) $\\displaystyle\\int 4x\\sin x\\,dx$

(c) $\\displaystyle\\int x\\cos 2x\\,dx$

(d) $\\displaystyle\\int x\\sqrt{x+1}\\,dx$

(e) $\\displaystyle\\int \\frac{x}{e^{3x}}\\,dx$

(f) $\\displaystyle\\int x\\sec^2 x\\,dx$

## Question 10 — Integration by Parts Twice

Find:

(a) $\\displaystyle\\int x^2\\sin x\\,dx$

(b) $\\displaystyle\\int x^2 e^{3x}\\,dx$

(c) $\\displaystyle\\int e^{-x}\\cos 2x\\,dx$

## Question 11 — Integration by Parts with Logarithms

Find:

(a) $\\displaystyle\\int \\ln 2x\\,dx$

(b) $\\displaystyle\\int 3x\\ln x\\,dx$

(c) $\\displaystyle\\int (\\ln x)^2\\,dx$

## Question 12 — Integration by Trigonometric Identities

(a) $\\displaystyle\\int 3\\sin^2 x\\,dx$

(b) $\\displaystyle\\int 4\\cos^2 x\\,dx$

(c) $\\displaystyle\\int 3\\sin x\\cos x\\,dx$

(d) $\\displaystyle\\int (2-3\\sin x)^2\\,dx$

(e) $\\displaystyle\\int (1-\\cos 2x)^2\\,dx$

(f) $\\displaystyle\\int 2\\tan^2 x\\,dx$

(g) $\\displaystyle\\int 5\\cot^2 x\\,dx$

(h) $\\displaystyle\\int (2\\tan x - \\cot x)^2\\,dx$

(i) $\\displaystyle\\int \\frac{4\\sin x}{\\cos^2 x}\\,dx$

(j) $\\displaystyle\\int \\frac{\\cos x}{3\\sin^2 x}\\,dx$

## All Integration Methods — Mixed Practice

Integrate with respect to $x$ unless stated otherwise.

1\. $(4x+5)^{\\frac{1}{2}}$

2\. $\\dfrac{1}{4x+5}$

3\. $\\left(1-\\dfrac{1}{x}\\right)^2$

4\. $\\cos x\\sin x$

5\. $\\tan 3x$

6\. $x\\sin 3x$

7\. $\\dfrac{1+x}{x^{\\frac{1}{2}}}$

8\. $\\dfrac{x}{1+x}$

9\. $\\sin x\\cos^4 x$

10\. $3\\ln x$

11\. $\\dfrac{x+2}{x(x-1)}$

12\. $\\dfrac{\\sec^2 x}{(1+\\tan x)^3}$

13\. $\\sin^2 2x$

14\. $\\dfrac{x^2}{x-2}$

15\. $(\\sin x + 2\\cos x)^2$

16\. $x^2 e^{4x}$

17\. $\\dfrac{1}{x^2-4}$

18\. $\\dfrac{x}{9x^2+1}$

19\. $(1-x^{-2})^2$

20\. $(2-3x)^{-2}$

21\. $(4-5x)^{-\\frac{1}{2}}$

22\. $\\cot 3x$

23\. $\\text{cosec}\\,2x\\cot 2x$

24\. $\\cot^2 3x$

25\. $x\\cos 5x$

26\. $\\dfrac{x}{(x-1)^{\\frac{1}{2}}}$

27\. $x^2 e^{-x}$

28\. $\\cos 2x\\sin x$

29\. $\\sin 2x\\cos x$

30\. $\\tan 2x\\sec 2x$

31\. $\\dfrac{(x+1)^2}{x^2+1}$

32\. $\\dfrac{2}{(x-2)(x-4)}$

33\. $\\dfrac{1}{x^2(x-1)}$

34\. $\\text{cosec}^2 2x + 1$

35\. $\\dfrac{x+4}{x-4}$

36\. $\\dfrac{1}{x(x^2-1)}$

37\. $\\dfrac{x^2}{x^3+1}$

38\. $(e^x + x)^2$

39\. $x^3\\ln x$

40\. $x^3 e^{x^2}$

41\. Use the identity $\\cos^2 x + \\sin^2 x \\equiv 1$ and the substitution $\\cos x = u$ to find $\\displaystyle\\int \\sin^3 x\\,dx$.

42\. Find $\\displaystyle\\int \\cos^3 x\\,dx$ and $\\displaystyle\\int \\sin^5 x\\,dx$.

43\. Use the identity $\\sec^2 x \\equiv \\tan^2 x + 1$ and the substitution $\\tan x = u$ to find $\\displaystyle\\int \\tan^4 x\\,dx$.

44\. Find (a) $\\displaystyle\\int \\sec^4 x\\,dx$ and (b) $\\displaystyle\\int \\cot^4 x\\,dx$.

45\. Use the identity $\\sin(A+B) + \\sin(A-B) \\equiv 2\\sin A\\cos B$ to find: (a) $\\displaystyle\\int 2\\sin 6x\\cos 4x\\,dx$ (b) $\\displaystyle\\int \\sin x\\cos\\tfrac{1}{2}x\\,dx$

46\. Evaluate $\\displaystyle\\int\_0^1 \\frac{x+9}{(x+2)(3-2x)}\\,dx$.

47\. Use the substitution $x = 3\\sin t$ to show that $\\displaystyle\\int\_0^3 x^2(9-x^2)^{\\frac{1}{2}}\\,dx = \\tfrac{81}{16}\\pi$.

48\. Evaluate $\\displaystyle\\int\_{\\pi/6}^{\\pi} \\sec^3 x\\tan x\\,dx$.

49\. Evaluate $\\displaystyle\\int\_3^4 \\frac{2x+4}{(x-2)(x^2+4)}\\,dx$.

50\. Evaluate $\\displaystyle\\int\_1^2 \\frac{x}{(1+x^2)}\\,dx$.