Integrals

Comprehensive Question Bank & Study Notes

Multiple Choice Questions

1 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1845
If $\int \frac{3ax}{b^2 + c^2x^2} dx = A \log |b^2 + c^2x^2| + K$, then the value of $A$ is
(A) $3a$
(B) $\frac{3a}{2b^2}$
(C) $\frac{3a}{b^2c^2}$
(D) $\frac{3a}{2c^2}$
MEDIUM
2 Knowledge REMEMBER MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1705
$\int\frac{1}{\sqrt{1+\cos 2x}}dx$ is equal to:
(A) $\log|\cos x|+C$
(B) $\frac{1}{\sqrt{2}}\log|\sec x-\tan x|+C$
(C) $\frac{1}{\sqrt{2}}\log|\sec x+\tan x|+C$
(D) $\log|\sin 2x|+C$
EASY
3 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1704
$\int\frac{dx}{\sqrt{25-16x^{2}}}$ is equal to:
(A) $\frac{1}{5}\sin^{-1}4x+C$
(B) $\frac{1}{25}\sin^{-1}16x+C$
(C) $\frac{1}{4}\sin^{-1}\frac{4x}{5}+C$
(D) $\frac{1}{16}\sin^{-1}\frac{4x}{5}+C$
MEDIUM
4 Competency APPLY MCQ_SINGLE 1 Mark(s) 2026 AISSCE(Board Exam)
#1703
$\int\frac{dx}{2^{x}+2^{-x}}$ is equal to:
(A) $\tan^{-1}(2^{x})+C$
(B) $\tan^{-1}(2^{-x})+C$
(C) $\frac{\tan^{-1}(2^{x})}{\log 2}+C$
(D) $(\log 2)\tan^{-1}(2^{x})+C$
MEDIUM
5 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#628
\(\int_{-1}^{1} \frac{|x|}{x} \, dx, x \ne 0 \text{ is equal to}\)
(A) \(-1\)
(B) 0
(C) 1
(D) 2
MEDIUM
6 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#627
If
\(
\int \frac{2^\frac{1}{x}}{x^2} dx = k \cdot 2^{\frac{1}{x}} + C,
\)
then \(k\) is equal to
(A) \(\dfrac{-1}{\log 2}\)
(B) \(-\log 2\)
(C) -1
(D) \(\dfrac{1}{2}\)
MEDIUM
7 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#626
\(\int\frac{\cos 2x-\cos 2\alpha}{\cos x-\cos \alpha}dx\) is equal to:
(A) \(2(\sin x+x\cos\alpha)+C\)
(B) \(2(\sin x-x\cos\alpha)+C\)
(C) \(2(\sin x+2x\cos\alpha)+C\)
(D) \(2(\sin x+\sin\alpha)+C\)
MEDIUM
8 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#625
\(\int\sqrt{1+\sin x}dx\) is equal to :
(A) \(2(-\sin\frac{x}{2}+\cos\frac{x}{2})+C\)
(B) \(2(\sin\frac{x}{2}-\cos\frac{x}{2})+C\)
(C) \(-2(\sin\frac{x}{2}+\cos\frac{x}{2})+C\)
(D) \(2(\sin\frac{x}{2}+\cos\frac{x}{2})+C\)
MEDIUM
9 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#624
\(\int\frac{x+5}{(x+6)^{2}}e^{x}dx\) is equal to:
(A) \(\log(x+6)+C\)
(B) \(e^{x}+C\)
(C) \(\frac{e^{x}}{x+6}+C\)
(D) \(\frac{-1}{(x+6)^{2}}+C\)
MEDIUM
10 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#623
For a function \(f(x)\) which of the following holds true?
(A) \(\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx\)
(B) \(\int_{-a}^{a}f(x)dx=0,\) if f is an even function
(C) \(\int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx,\) if f is an odd function
(D) \(\int_{0}^{2a}f(x)dx=\int_{0}^{a}f(x)dx-\int_{0}^{a}f(2a+x)dx\)
MEDIUM
11 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#622
\(\int\frac{e^{x}}{\sqrt{4-e^{2x}}}dx\) is equal to:
(A) \(\frac{1}{2}\cos^{-1}(e^{x})+C\)
(B) \(\frac{1}{2}\sin^{-1}(e^{x})+C\)
(C) \(\frac{e^{x}}{2}+C\)
(D) \(\sin^{-1}(\frac{e^{x}}{2})+C\)
MEDIUM
12 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#621
\(\int\frac{1}{x(\log~x)^{2}}dx\) is equal to:
(A) \(2~log(log~x)+c\)
(B) \(-\frac{1}{log~x}+c\)
(C) \(\frac{(log~x)^{3}}{3}+c\)
(D) \(\frac{3}{(log~x)^{3}}+c\)
MEDIUM
13 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#808
$\int\frac{2\cos 2x-1}{1+2\sin x}dx$ is equal to:
(A) $x-2\cos x+C$
(B) $x+2\cos x+C$
(C) $-x-2\cos x+C$
(D) $-x+2\cos x+C$
MEDIUM
14 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#807
$\int 2^{x+2}dx$ is equal to :
(A) $2^{x+2}+C$
(B) $2^{x+2}\log 2+C$
(C) $\frac{2^{x+2}}{\log 2}+C$
(D) $2\cdot\frac{2^{x}}{\log 2}+C$
MEDIUM
15 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#806
9. If $\int_{0}^{a}3x^{2}dx=8$, then the value of 'a' is :
(A) 2
(B) 4
(C) 8
(D) 10
MEDIUM
16 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#805
8. $\int e^{5\log x}dx$ is equal to :
(A) $\frac{x^{5}}{5}+C$
(B) $\frac{x^{6}}{6}+C$
(C) $5x^{4}+C$
(D) $6x^{5}+C$
MEDIUM
17 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#804
\int_{0}^{\frac{\pi}{6}}sec^{2}(x-\frac{\pi}{6})dx is equal to :
(A) \frac{1}{\sqrt{3}}
(B) -\frac{1}{\sqrt{3}}
(C) \sqrt{3}
(D) -\sqrt{3}
MEDIUM
18 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2023
#803
If \frac{d}{dx}(f(x))=log\~x, then f(x) equals :
(A) -\frac{1}{x}+C
(B) x(log\~x-1)+C
(C) x(log\~x+x)+C
(D) \frac{1}{x}+C
MEDIUM
19 Competency APPLY MCQ_SINGLE 1 Mark(s) 2023
#802
∫ from -1 to 1 [|x-2| / (x-2)] dx, x≠2 is equal to
(A) 1
(B) -1
(C) 2
(D) -2
MEDIUM
20 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2023
#801
∫ [sec x / (sec x - tan x)] dx equals
(A) sec x - tan x + c
(B) sec x + tan x + c
(C) tan x - sec x + c
(D) -(sec x + tan x) + c
MEDIUM

Very Short Answer (2 Mark)

1 Knowledge UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1338
Find: $\int\frac{1}{5+4x-x^{2}}dx$
MEDIUM
2 Knowledge UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1337
Find: $\int cos^{3}x~e^{log~sin~x}dx$
MEDIUM
3 Knowledge APPLY VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1316
Find: $\int\frac{1}{x(x^{2}-1)}dx.$
MEDIUM
4 Knowledge UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1291
Find: $\int x\sqrt{1+2x}dx$
MEDIUM
5 Knowledge UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1271
Find: $\int\frac{e^{4x}-1}{e^{4x}+1}dx$
MEDIUM
6 Knowledge UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1248
Given $\frac{d}{dx}F(x)=\frac{1}{\sqrt{2x-x^{2}}}$ and $F(1)=0$, find $F(x)$.
MEDIUM
7 Knowledge APPLY VSA 1 Mark(s)
#1492
Evaluate the following integral $\int x^2 dx$

Short Answer (3 Marks)

1 Knowledge SA 3 Mark(s) 2026 AISSCE(Board Exam)
#1783
Find: $\int\frac{x-\sin x}{1-\cos x}dx$
EASY
2 Knowledge SA 3 Mark(s) 2026 AISSCE(Board Exam)
#1782
Find: $\int\frac{x+3}{x^{2}+4x+5}dx$
EASY
3 Knowledge SA 3 Mark(s) 2026 AISSCE(Board Exam)
#1781
Find: $\int\frac{\cos x}{(2+\sin x)(4+\sin x)}dx$
EASY
4 Knowledge SA 3 Mark(s) 2026 AISSCE(Board Exam)
#1780
Find: $\int\tan^{-1}(\frac{1-x}{1+x})dx$
EASY
5 Knowledge SA 3 Mark(s) 2026 AISSCE(Board Exam)
#1779
Find: $\int\frac{dx}{x^{1/2}+x^{1/3}}$
EASY
6 Knowledge SA 3 Mark(s) 2026 AISSCE(Board Exam)
#1778
If $\frac{d}{dx}(F(x))=\frac{1}{e^{x}+1}$ then find $F(x)$ given that $F(0)=\log\frac{1}{2}$
EASY
7 Knowledge SA 3 Mark(s) 2026 AISSCE(Board Exam)
#1777
Find: $\int\frac{x+2}{\sqrt{9x-x^{2}}}dx$
EASY
8 Knowledge REMEMBER SA 3 Mark(s) 2026 AISSCE(Board Exam)
#1503
$\text{Find: }\int \frac{x^{2}}{(x^{2}+9)(x^{2}+16)}\,dx.$
EASY
9 Knowledge REMEMBER SA 3 Mark(s) 2026 AISSCE(Board Exam)
#1502
$\text{Find }\int \sqrt{\frac{x+2}{x-2}}\,dx.$
EASY
10 Knowledge UNDERSTAND SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1429
Find: $\int\frac{2x}{(x^{2}+3)(x^{2}-5)}dx$.
MEDIUM
11 Knowledge UNDERSTAND SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1392
Find: $\int\frac{1}{x}\sqrt{\frac{x+a}{x-a}}dx$.
HARD
12 Knowledge APPLY SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1364
Find: $\int\frac{x+\sin x}{1+\cos x}dx$.
MEDIUM
13 Knowledge UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1344
Find: $\int\frac{2x+1}{(x+1)^{2}(x-1)}dx$
HARD
14 Knowledge UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1320
Find: $\int\frac{3x+5}{\sqrt{x^{2}+2x+4}}dx$
MEDIUM
15 Knowledge UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1319
Find: $\int e^{x}[\frac{1}{(1+x^{2})^{\frac{3}{2}}}+\frac{x}{\sqrt{1+x^{2}}}]dx$
HARD
16 Knowledge UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1299
Find: $\int\frac{x^{2}}{(x^{2}+4)(x^{2}+9)}dx$
MEDIUM
17 Knowledge UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1281
Find: $\int x^{2}\cdot sin^{-1}(x^{3/2})dx$
MEDIUM
18 Knowledge APPLY SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1276
Find: $\int\frac{1}{x[(log~x)^{2}-3~log~x-4]}dx$
MEDIUM
19 Knowledge UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1256
Find: $\int\frac{2+sin~2x}{1+cos~2x}e^{x}dx$
MEDIUM
20 Knowledge UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1255
Find: $\int\frac{x^{2}+1}{(x^{2}+2)(x^{2}+4)}dx$
MEDIUM
21 Knowledge APPLY SA 3 Mark(s) 2023
#905
Find: $\int x^{2}\log(x^{2}+1)dx$
MEDIUM
22 Knowledge APPLY SA 3 Mark(s) 2023
#904
Find: $\int\frac{\cos x}{\sin 3x}dx$
MEDIUM
23 Knowledge APPLY SA 3 Mark(s) 2023
#903
Evaluate $\int_{1}^{e}\frac{1}{\sqrt{4x^{2}-(x\log x)^{2}}}dx$
MEDIUM
24 Knowledge APPLY SA 3 Mark(s) 2023
#902
Evaluate: $\int_{1}^{3}\frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}}dx$
MEDIUM
25 Knowledge APPLY SA 3 Mark(s) 2023
#901
28. (b) OR: Evaluate: $\int_{0}^{\pi/2}\sqrt{\sin x}\cos^{5}x~dx$
HARD
26 Knowledge APPLY SA 3 Mark(s) 2023
#900
28. (a) Find: $\int\frac{e^{x}}{\sqrt{5-4e^{x}-e^{2x}}}dx$
MEDIUM
27 Knowledge APPLY SA 3 Mark(s) 2023
#899
27. (b) OR: Evaluate: $\int_{-2}^{2}\frac{x^{2}}{1+5^{x}}dx$
MEDIUM
28 Knowledge APPLY SA 3 Mark(s) 2023
#898
27. (a) Evaluate: $\int_{\pi/4}^{\pi/2}e^{2x}(\frac{1-\sin 2x}{1-\cos 2x})dx$
HARD
29 Knowledge APPLY SA 3 Mark(s) 2023
#897
26. Find: $\int\frac{x^{2}+x+1}{(x+1)^{2}(x+2)}dx$
MEDIUM
30 Knowledge APPLY SA 3 Mark(s) 2023
#896
Evaluate : $\int_{0}^{\frac{\pi}{2}}e^{x }\sin x~dx$.
OR
Find: $\int\frac{1}{\cos(x-a)\cos(x-b)}dx$
MEDIUM
31 Knowledge APPLY SA 3 Mark(s) 2023
#895
Find:$\int \frac{1}{\sqrt{x}(\sqrt{x}+1)(\sqrt{x}+2)} \, dx$
MEDIUM
32 Knowledge APPLY SA 3 Mark(s) 2023
#894
Evaluate $\int_{0}^{\frac{\pi}{2}}[\log(\sin~x)-\log(2\cos~x)]dx.$
MEDIUM
33 Knowledge APPLY SA 3 Mark(s) 2023
#892
(a) Evaluate: $\int_0^{2\pi} \frac{1}{1 + e^{\sin x}} dx $
OR
(b) Find: $\int \frac{x⁴} { ((x-1)(x²+1))}dx.$
HARD

Long Answer (5 Marks)

1 Competency APPLY LA 5 Mark(s) 2026 AISSCE(Board Exam)
#1817
Find: $\int\frac{x}{(x-1)(x^{2}+4)}dx$
MEDIUM
2 Knowledge UNDERSTAND LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1481
Find: $\int\frac{x^{2}+1}{(x^{2}+2)(2x^{2}+1)}dx$
HARD
3 Knowledge UNDERSTAND LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1459
Find: $\int\frac{x^{2}+1}{(x-1)^{2}(x+3)}dx$.
HARD
4 Knowledge UNDERSTAND LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1417
Find: $\int\frac{\cos x}{(4+\sin^{2}x)(5-4\cos^{2}x)}dx$.
HARD
5 Knowledge LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1394
Find: $\int\frac{x^{2}+x+1}{(x+2)(x^{2}+1)}dx$.
6 Knowledge APPLY LA 5 Mark(s) 2023
#893
Evaluate: $\int_ 0 ^{π/2} [\sin 2x \tan⁻¹(\sin x)] dx$
MEDIUM