A-Level Maths | Integration
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 $\displaystyle\int_1^2 \frac{x}{(1+x^2)}\,dx$.