1.If \[I=\int_{0}^{2\pi}e^{x/2}\sin\left(\frac{x}{2}+\frac{\pi}{4}\right)dx\]
then I equals
a) \[\pi\]
b) 0
c) \[-\pi/2\]
d) \[2\pi\]
Explanation:
2. If \[I=\int_{0}^{2\pi}e^{x}\cos\left(\frac{x}{2}+\frac{\pi}{4}\right)dx\]
then I equals
a) \[\frac{-3}{5}\left(e^{2\pi}-1\right)\]
b) \[\frac{-3\sqrt{2}}{5}\left(e^{2\pi}-1\right)\]
c) \[\frac{-3\sqrt{2}}{5}\left(e^{2\pi}+1\right)\]
d) 0
Explanation:
3.Let a, b, c, > 0 and b > c, If
\[I=e^{a}\int_{b}^{c} \frac{x^{4}+1}{x^{6}+1}dx\]
then
a) \[I<\left(\frac{c}{b}\right)^{a}\]
b) \[I>e^{ab}\]
c) \[I< e^{ac}\]
d) \[I< e^{a\left(c-b\right)}\]
Explanation:
4. Let \[I=4\int_{0}^{\pi/4}\frac{x^{2}\left(\sin 2x-\cos 2x\right)}{\left(\sin x+\cos x\right)^{2}\cos^{2} x} dx\]
a) \[\frac{\pi^{2}}{4}-\pi\log\left(2\right)\]
b) \[\frac{\pi^{2}}{2}+\frac{1}{2}\pi\log\left(2\right)\]
c) \[\pi^{2}-\frac{3}{4}\pi\log\left(2\right)\]
d) \[\frac{\pi^{2}}{16}+\frac{1}{4}\pi\log\left(2\right)\]
Explanation:
5. If \[I=\int_{\pi/3}^{2\pi/3} \frac{x}{1+\sin x}dx\]
then I equal to
a) \[2\pi\]
b) \[\left(2+\sqrt{3}\right)\pi\]
c) \[\left(2-\sqrt{3}\right)\pi\]
d) 0
Explanation:
6. If \[I=\int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\tan x} }dx\]
then I equals
a) \[\frac{\pi}{12}\]
b) \[\frac{\pi}{6}\]
c) \[\frac{\pi}{4}\]
d) \[\frac{\pi}{3}\]
Explanation:
7. If \[I_{1}=\int_{0}^{\pi/1} f\left(\sin 2x\right)\sin x dx\]
and \[I_{2}=\int_{0}^{\pi/4} f\left(\cos 2x\right)\cos x dx\]
then \[\frac{I_{1}}{I_{2}}\] equals
a) 1
b) \[1/\sqrt{2}\]
c) \[\sqrt{2}\]
d) 2
Explanation:
8. If \[\int_{0}^{1}\frac{\sin t}{1+t}dt=\alpha\]
then value of
\[I=\int_{4\pi-2}^{4\pi}\frac{\sin\left(x/2\right)}{4\pi+2-x}dx\]
is
a) \[\alpha/2\]
b) \[-\alpha\]
c) \[-\alpha/2\]
d) \[\alpha\]
Explanation:
9. If \[I=\int_{-\pi}^{\pi}\frac{2x\left(1+\sin x\right)}{1+\cos^{2} x}dx\]
then I equals
a) \[-\sqrt{2}\pi^{2}\]
b) \[\pi^{2}\]
c) \[\frac{\pi^{2}}{\sqrt{2}}\]
d) \[-\frac{\pi^{2}}{\sqrt{2}}\]
Explanation:
10. If \[h\left(x\right)=\int_{1}^{x} \sin^{4}t dt\]
then h(x + \[\pi\] ) equals
a) h(x) + h( \[\pi\] )
b) h(x) h( \[\pi\] )
c) h(x) - h( \[\pi\] )
d) h(x)/h( \[\pi\] )
Explanation:
