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#636
Mathematics
Definite Integrals
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
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) \(2\int_{0}^{a}f(x)dx\)
Key:
Sol:
Sol:
**Correct Option if MCQ:** D
**Reasoning:**
* Let \(I = \int_{0}^{2a}f(x)dx\).
* Using the property \(\int_{0}^{na}f(x)dx = n\int_{0}^{a}f(x)dx\) for \(f(x)=f(2a-x)\).
* Thus, \(I = 2\int_{0}^{a}f(x)dx\).
#635
Mathematics
Definite Integrals
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
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) \(\tan^{-1}e\)
Key: C
Sol:
Sol:
\[\frac{1}{e^{x}+e^{-x}} = \frac{1}{e^{x} + \frac{1}{e^{x}}} = \frac{1}{\frac{e^{2x} + 1}{e^x}} = \frac{e^x}{e^{2x} + 1}\]
The integral becomes:
\[I = \int_{0}^{1}\frac{e^x}{e^{2x} + 1}dx\]Now use Substitution
Let \(u = e^x\). Then, \(du = e^x dx\).
Change the limits of integration:
- Lower limit (\(x=0\)): \(u_1 = e^0 = 1\)
- Upper limit (\(x=1\)): \(u_2 = e^1 = e\)
The integral becomes:
\[I = \int_{1}^{e}\frac{du}{u^{2} + 1}\]Now evaluate and Apply Limits
\[I = \left[ \tan ^{-1}(u) \right]_{1}^{e}\] \[=\tan ^{-1}(e) - \frac{\pi}{4}\]
#634
Mathematics
Definite Integrals
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
\(\int_{0}^{\pi/2}\cos x\cdot e^{\sin x}dx\) is equal to:
(A) 0
(B) \(1-e\)
(C) \(e-1\)
(D) e
Key:
Sol:
Sol:
#633
Mathematics
Definite Integrals
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
\(\int_{a}^{b}f(x)dx\) 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) \(\int_{a}^{b}f((a-x)+(b-x))dx\)
Key:
Sol:
Sol:
#632
Mathematics
Definite Integrals
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
\(\int_{0}^{\pi/2}\frac{\sin~x-\cos~x}{1+\sin~x~\cos~x}dx\) is equal to:
(A) \(\pi\)
(B) Zero (0)
(C) \(\int_{0}^{\pi/2}\frac{2~\sin~x}{1+\sin~x~\cos~x}dx\)
(D) \(\frac{\pi^{2}}{4}\)
Key: B
Sol:
Sol:
#631
Mathematics
Definite Integrals
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
\(\int_{-a}^{a}f(x)dx=0,\) if :
(A) \(f(-x)=f(x)\)
(B) \(f(-x)=-f(x)\)
(C) \(f(a-x)=f(x)\)
(D) \(f(a-x)=-f(x)\)
Key:
Sol:
Sol:
#630
Mathematics
Definite Integrals
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
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) \(\frac{\pi}{18}\)
Key:
Sol:
Sol:
#629
Mathematics
Definite Integrals
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The value of \(\int_{-1}^{1}x|x|dx\) is:
(A) \(\frac{1}{6}\)
(B) \(\frac{1}{3}\)
(C) \(-\frac{1}{6}\)
(D) 0
Key:
Sol:
Sol: