1.Let f, g be continuously differentiable
functions from R+ to R+ then the value of
\[\int_{a}^{b} \frac{f\left(x\right)g'\left(x\right)-f'\left(x\right)}{f\left(x\right)+e^{g\left(x\right)}}dx\]
is equal to
a) \[\log\left(\frac{e^{-g\left(b\right)}f\left(b\right)+1}{e^{-g\left(a\right)}f\left(a\right)+1}\right)\]
b) \[\log\left(\frac{e^{-g\left(a\right)}f\left(a\right)+1}{e^{-g\left(b\right)}f\left(b\right)+1}\right)\]
c) \[\log\left(\frac{e^{-g\left(a\right)}f\left(b\right)+1}{e^{-g\left(a\right)}f\left(a\right)+1}\right)\]
d) \[\log\left(\frac{e^{-g\left(b\right)}f\left(a\right)+1}{e^{-g\left(a\right)}f\left(b\right)+1}\right)\]
Explanation:
2. The value of \[I= \int_{0}^{\pi} \frac{x\sin x}{\cos 4x+4\cos 2x+11}dx\]
is
a) \[\frac{\pi}{32}\left(\pi-\log\left(1+2\sqrt{2}\right)\right)\]
b) \[\frac{\pi}{32\sqrt{2}}\left(\pi-\log\left(3-2\sqrt{2}\right)\right)\]
c) \[\frac{\pi}{16}\left(\pi-\log\left(3+2\sqrt{2}\right)\right)\]
d) \[\frac{\pi}{16}\left(\pi-\log\left(3-2\sqrt{2}\right)\right)\]
Explanation:
3.If \[I=\int_{-\pi/6}^{\pi/6} \frac{\pi+4x^{5}}{1-\sin\left(\mid x \mid+\pi/6\right)}dx\]
then I equals
a) \[4\pi\]
b) \[2\pi+1/\sqrt{3}\]
c) \[2\pi-\sqrt{3}\]
d) \[4\pi+\sqrt{3}-1\]
Explanation:
4. If \[I=\int_{-3}^{2} \left(\mid x+1\mid+\mid x+2\mid+\mid x-1\mid\right)dx\]
then I equals
a) \[\frac{31}{2}\]
b) \[\frac{35}{2}\]
c) \[\frac{37}{2}\]
d) \[\frac{39}{2}\]
Explanation:
5. If \[I=\int_{0}^{1.7} \left[x^{2}\right]dx\]
then I equals
a) \[2.4+\sqrt{2}\]
b) \[2.4-\sqrt{2}\]
c) \[3.4+\sqrt{2}\]
d) \[2.4-1/\sqrt{2}\]
Explanation:
6.The value of \[I=\int_{-2}^{0} \left[x^{3}+3x^{2}+3x+\left(x+1\right)\cos\left(x+1\right)\right]dx\]
is
a) -4
b) -3
c) -2
d) -1
Explanation:
7. If \[I=\int_{0}^{\pi} e^{\mid\left(1/2\right)\cos x \mid}\left\{2\sin\left(\frac{1}{2}\cos x\right)+3\cos\left(\frac{1}{2}\cos x\right)\right\}\sin x dx\]
then I equals
a) \[7\sqrt{e}\cos \left(1/2\right)\]
b) \[7\sqrt{e}\left[\cos \left(1/2\right)-\sin\left(1/2\right)\right]\]
c) 0
d) \[6\left(\sqrt{e}\left(\cos\frac{1}{2}+\sin\frac{1}{2}\right)-1\right)\]
Explanation:
8. If \[I=\int_{-2}^{2} \mid 1-x^{4}\mid dx\]
then I equals
a) 6
b) 8
c) 12
d) 21
Explanation:
9. The value of \[\int_{-1}^{2} \mid \left[x\right]-\left\{x\right\}\mid dx\]
, where [x]
is the greatest integer less then or equal to x and {x} is the
fractional part of x is
a) 7/2
b) 5/2
c) 1/2
d) 3/2
Explanation:
10. If \[I=\int_{2}^{3}\frac{2x^{5}+x^{4}-2x^{3}+2x^{2}+1}{\left(x^{2}+1\right)\left(x^{4}-1\right)}dx\]
then I equals
a) \[\frac{1}{2}\log 6+\frac{1}{10}\]
b) \[\frac{1}{2}\log 6-\frac{1}{10}\]
c) \[\frac{1}{2}\log 3-\frac{1}{10}\]
d) \[\frac{1}{2}\log 2+\frac{1}{10}\]
Explanation:
