1.If \[x=\int_{c^{2}}^{\tan t} \tan^{-1} z dz ,y=\int_{n}^{\sqrt{t}} \frac{\cos\left(z^{2}\right)}{z}dz\]
than
\[\frac{dy}{dx}\] is equal to
a) \[\frac{\tan t}{2 t}\]
b) \[\frac{\cos^{2} t}{ t^{2}}\]
c) \[\frac{\cos^{3} t}{ 2t^{2}}\]
d) \[\frac{\tan t^{2}}{ 2t^{2}}\]
Explanation:
2.If \[\int_{\log 2}^{x} \frac{dx}{\sqrt{e^{x}-1}}=\frac{\pi}{6}\]
the value of x is
a) 4
b) \[\log 2\]
c) \[\log 4\]
d) \[\log 8\]
Explanation:
3.If \[I=\int_{1/ 2}^{1} \frac{dx}{x^{4}\sqrt{1-x^{2}}}\]
then I is equal to
a) 2
b) 2/3
c) \[2/\sqrt{3}\]
d) \[\sqrt{3}\]
Explanation:
4. The value of \[\int_{-4}^{-5} e^{\left(x+5\right)^{2}}dx+3\int_{1/3}^{2/3} e^{9\left(x-2/3\right)^{2}}dx\]
is
a) 2/5
b) 1/5
c) 1/2
d) none of these
Explanation:
5. If \[f\left(x\right)= A\sin \left(\pi x/2\right)+B,f'\left(1/2\right)=\sqrt{2}\]
and
\[\int_{0}^{1}f\left(x\right) dx= \frac{2A}{\pi}\] , then the constants A and B are (resp.)
a) \[\pi/2,\pi/2\]
b) \[2/\pi,3\pi\]
c) \[0,-4\pi\]
d) \[4/\pi,0\]
Explanation:
6.\[\int_{\pi/2}^{3\pi/2}\] [2 sin x] dx is equal to ([x] denotes the greatest
integer function)
a) \[-\pi\]
b) 0
c) \[-\pi/2\]
d) \[\pi/2\]
Explanation:
7. If \[f\left(x\right)=\begin{cases}e^{\cos x}\sin x & for\mid x \mid \leq 2\\2 & other wise \end{cases}\]
then \[\int_{-2}^{3}f\left(x\right) dx\] is equal to
a) 0
b) 1
c) 2
d) 3
Explanation:
8. The value of \[\int_{e^{-1}}^{e^{2}}\mid\frac{\log x}{x} \mid dx\] is
a) 3/2
b) 5/2
c) 3
d) 5
Explanation:
9. The value of\[\int_{-\pi}^{\pi}\frac{\cos^{2} x}{1+a^{x}} dx ,a>0\] is
a) \[\pi/2\]
b) \[a\pi\]
c) \[\pi\]
d) \[2\pi\]
Explanation:
10. The integral \[\int_{-1/2}^{1/2}\left(\left[x\right]+\log\frac{1+x}{1-x}\right)dx\]
equals
a) -1/2
b) 0
c) 1
d) -2 log 2
Explanation:
