Definite Integrals

Comprehensive Question Bank & Study Notes

Multiple Choice Questions

1 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#636
If \(f(2a-x)=f(x)\), then \(\int_{0}^{2a}f(x)dx\) is
(A) \(\int_{0}^{2a}f(\frac{x}{2})dx\)
(B) \(\int_{0}^{a}f(x)dx\)
(C) \(2\int_{a}^{0}f(x)dx\)
(D) \(2\int_{0}^{a}f(x)dx\)
MEDIUM
2 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#635
The value of \(\int_{0}^{1}\frac{dx}{e^{x}+e^{-x}}\) is:
(A) \(-\frac{\pi}{4}\)
(B) \(\frac{\pi}{4}\)
(C) \(\tan^{-1}e-\frac{\pi}{4}\)
(D) \(\tan^{-1}e\)
MEDIUM
3 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025 AISSCE(Board Exam)
#634
\(\int_{0}^{\pi/2}\cos x\cdot e^{\sin x}dx\) is equal to:
(A) 0
(B) \(1-e\)
(C) \(e-1\)
(D) e
MEDIUM
4 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#633
\(\int_{a}^{b}f(x)dx\) is equal to:
(A) \(\int_{a}^{b}f(a-x)dx\)
(B) \(\int_{a}^{b}f(a+b-x)dx\)
(C) \(\int_{a}^{b}f(x-(a+b))dx\)
(D) \(\int_{a}^{b}f((a-x)+(b-x))dx\)
MEDIUM
5 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#632
\(\int_{0}^{\pi/2}\frac{\sin~x-\cos~x}{1+\sin~x~\cos~x}dx\) is equal to:
(A) \(\pi\)
(B) Zero (0)
(C) \(\int_{0}^{\pi/2}\frac{2~\sin~x}{1+\sin~x~\cos~x}dx\)
(D) \(\frac{\pi^{2}}{4}\)
MEDIUM
6 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#631
\(\int_{-a}^{a}f(x)dx=0,\) if :
(A) \(f(-x)=f(x)\)
(B) \(f(-x)=-f(x)\)
(C) \(f(a-x)=f(x)\)
(D) \(f(a-x)=-f(x)\)
MEDIUM
7 Knowledge UNDERSTAND MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#630
The value of \(\int_{0}^{3}\frac{dx}{\sqrt{9-x^{2}}}\) is:
(A) \(\frac{\pi}{6}\)
(B) \(\frac{\pi}{4}\)
(C) \(\frac{\pi}{2}\)
(D) \(\frac{\pi}{18}\)
MEDIUM
8 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2024 AISSCE(Board Exam)
#629
The value of \(\int_{-1}^{1}x|x|dx\) is:
(A) \(\frac{1}{6}\)
(B) \(\frac{1}{3}\)
(C) \(-\frac{1}{6}\)
(D) 0
MEDIUM

Very Short Answer (2 Mark)

1 Knowledge UNDERSTAND VSA 2 Mark(s) 2025 AISSCE(Board Exam)
#1379
Evaluate: $\int_{0}^{\frac{\pi}{4}}\sqrt{1+\sin 2x}dx$.
MEDIUM
2 Knowledge APPLY VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1292
Evaluate: $\int_{0}^{\frac{\pi^{2}}{4}}\frac{sin\sqrt{x}}{\sqrt{x}}dx$
MEDIUM
3 Knowledge UNDERSTAND VSA 2 Mark(s) 2024 AISSCE(Board Exam)
#1247
Evaluate: $\int_{0}^{\pi/2}sin~2x~cos~3x~dx$
MEDIUM

Short Answer (3 Marks)

1 Competency APPLY SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1430
Evaluate: $\int_{1}^{4}(|x-2|+|x-4|)dx$.
MEDIUM
2 Knowledge UNDERSTAND SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1410
Evaluate: $\int_{\pi/2}^{\pi}e^{x}\left(\frac{1-\sin x}{1-\cos x}\right)dx$.
HARD
3 Knowledge SA 3 Mark(s) 2025 AISSCE(Board Exam)
#1365
Evaluate: $\int_{0}^{\frac{\pi}{4}}\frac{dx}{\cos^{3}x\sqrt{2\sin 2x}}$.
4 Knowledge REMEMBER SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1343
Evaluate : $\int_{0}^{\pi}\frac{e^{cos~x}}{e^{cos~x}+e^{-cos~x}}d~x$
MEDIUM
5 Competency APPLY SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1318
Evaluate $\int_{0}^{\frac{\pi}{4}}\frac{x~dx}{1+cos~2x+sin~2x}$
MEDIUM
6 Knowledge UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1300
Evaluate: $\int_{1}^{3}(|x-1|+|x-2|+|x-3|)dx$
MEDIUM
7 Knowledge UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1275
Evaluate : $\int_{-2}^{2}\sqrt{\frac{2-x}{2+x}}dx$
HARD
8 Knowledge UNDERSTAND SA 3 Mark(s) 2024 AISSCE(Board Exam)
#1257
Evaluate: $\int_{0}^{\pi/4}\frac{1}{sin~x+cos~x}dx$
MEDIUM

Long Answer (5 Marks)

1 Knowledge UNDERSTAND LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1482
Evaluate: $\int_{0}^{\pi}\frac{x\tan x}{\sec x+\tan x}dx$
HARD
2 Knowledge UNDERSTAND LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1460
Evaluate: $\int_{0}^{\pi/2}\frac{x}{\sin x+\cos x}dx$.
HARD
3 Knowledge UNDERSTAND LA 5 Mark(s) 2025 AISSCE(Board Exam)
#1418
Evaluate: $\int_{0}^{\pi}\frac{dx}{a^{2}\cos^{2}x+b^{2}\sin^{2}x}$.
MEDIUM
4 Knowledge APPLY LA 5 Mark(s) 2024 AISSCE(Board Exam)
#1307
Evaluate: $\int_{0}^{\frac{\pi}{2}}sin~2x~tan^{-1}(sin~x)dx$
MEDIUM
5 Knowledge REMEMBER LA 5 Mark(s) 2024 AISSCE(Board Exam)
#1306
Evaluate: $\int_{0}^{\frac{\pi}{4}}\frac{sin~x+cos~x}{9+16~sin~2x}dx$
MEDIUM