1. If \[I=\int_{0}^{\pi/2}\cos^{n}x\sin^{n}x dx=\lambda\int_{0}^{\pi/2} sin^{n}x dx\]
then \[\lambda\] equals
a) \[2^{-n+1}\]
b) \[2^{-n-1}\]
c) \[2^{-n}\]
d) \[2^{-1}\]
Explanation:
2. If \[I=\int_{0}^{\pi} \sin ^{3}x\left(1+2\cos x\right)\left(1+\cos x\right)^{2} dx\]
then I equals
a) \[\frac{4}{3}\]
b) \[\frac{2}{3}\]
c) \[\frac{8}{3}\]
d) 2
Explanation:
3.If \[I=\int_{1/3}^{3} \frac{1}{x}\sin \left(\frac{1}{x}-x\right)dx\]
then I equals
a) \[\sqrt{3}/2\]
b) \[\pi+\sqrt{3}/2\]
c) 0
d) \[\pi-\sqrt{3}/2\]
Explanation:
4. If \[I=\int_{0}^{\pi/2} \frac{\sin 8 x\log \left(\cot x\right)}{\cos 2x})dx\]
then I equals
a) \[-\pi/2\]
b) \[\pi/3\]
c) - 1/3
d) 0
Explanation:
5. If \[I=\int_{1/e}^{e} \mid\log x \mid \frac{dx}{x^{2}}\]
then I equals
a) 2
b) 2/e
c) 2(1-1/e)
d) 0
Explanation:
6. If for \[ K\epsilon N,\]
\[\frac{\sin2kx}{\sin x}=2\left[\cos x+\cos3x+....+\cos\left(2k-1\right)x\right]\]
then value of \[I=\int_{0}^{\pi/2} \sin2kx\cot x dx\]
is
a) \[-\pi/2\]
b) 0
c) \[\pi/2\]
d) \[\pi\]
Explanation:
7.The natural number \[n\left(\leq 5\right)\] for which
\[I_{n}=\int_{0}^{1} e^{x}\left(x-1\right)^{n}dx=16-6e\]
is
a) 2
b) 3
c) 4
d) 5
Explanation:
8. If \[I=\int_{0}^{\pi/2} \frac{\sin^{3}x\cos x}{\sin^{4}x+\cos^{4}x}dx\]
then I equals
a) \[\pi/8\]
b) \[\pi/4\]
c) \[\pi/2\]
d) \[\pi\]
Explanation:
9. Let f : [0, 1] \[\rightarrow\] R be a continuous function
such that \[\int_{0}^{\pi} f\left(\sin x\right)dx=2018\]
then \[\int_{0}^{\pi}xf\left(\sin x\right)dx\]
is equal to
a) \[1009\pi\]
b) \[2016\pi\]
c) \[1008\pi\]
d) \[2017\pi\]
Explanation:
10. For a > 0, let \[I\left(a\right)=\frac{3}{2}\int_{0}^{a} \frac{\sqrt{x\left(x^{2}+x+1\right)}}{\sqrt{x+1}\sqrt{x^{4}+x^{2}+1}}dx\]
then I (a) is equal to
a) \[ln \left(\sqrt{a+1}+\sqrt{a}\right)\]
b) \[ln \left(\sqrt{a^{2}+1}+a\right)\]
c) \[ln \left(\sqrt{a^{3}+1}+\sqrt{a^{3}}\right)\]
d) \[2ln \left(\sqrt{a^{2}+1}+a\right)\]
Explanation:
