1.Let \[I=\int_{0}^{1} \] (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) dx,
then
a) I > 400
b) I > 700
c) I < 300
d) I < 200
Explanation:
2. If \[I=\int_{1}^{2} x^{-x}dx\]
then
a) \[I<\frac{3}{4}\]
b) \[I\geq\frac{3}{4}\]
c) I>1
d) I>2
Explanation:
3. If \[I=\int_{1/3}^{3} \sin^{11}\left(x-\frac{1}{x}\right)\frac{dx}{x}\]
then I is equal to
a) 0
b) sin 3 – sin (1/3)
c) \[\frac{1}{12}\left(\sin3-\sin\left(1/3\right)\right)^{5}\]
d) \[\frac{1}{6}\left(\sin3-\sin 1/3\right)^{4}\]
Explanation:
4. If \[I=\int_{0}^{\infty} \frac{\tan^{-1}x}{x^{2}+x+1}dx\]
then I is equal to
a) \[\frac{\pi^{2}}{4\sqrt{3}}\]
b) \[\frac{\pi^{2}}{6\sqrt{3}}\]
c) \[\frac{\pi^{2}}{4}\]
d) \[\frac{\pi^{2}}{2\sqrt{2}}\]
Explanation:
5. If \[I=\int_{0}^{\pi/4}\frac{\sin4x }{\sqrt{1+\sin^{2}x}}dx\]
then I is equal to
a) \[8\sqrt{\frac{3}{2}}\]
b) \[\frac{28}{3}-\frac{\pi}{3}\]
c) \[8\sqrt{\frac{3}{2}}-\frac{28}{3}\]
d) \[\frac{\pi^{2}}{16}\]
Explanation:
6. If \[I=\int_{-\pi/2}^{\pi/2}\frac{\sqrt{\cos^{2}x+\cos^{4}x }}{5^{x}+1}dx\]
then I is equal to
a) \[\frac{1}{4}ln5+\frac{\pi}{6}\]
b) \[\frac{1}{2}ln5+\frac{\pi}{6}\]
c) \[\frac{1}{2}+\frac{\pi}{6}\]
d) \[\frac{3}{2}+\frac{\pi}{6}\]
Explanation:
7. If \[I=\int_{0}^{2016}x\left(x-4\right)\left(x-8\right)\left(x-12\right)....\left(x-2016\right)dx\]
then I is equal to
a) 2016
b) 1008
c) 0
d) \[\left(10008\right)^{2}\]
Explanation:
8. If \[f(x)=\int_{1}^{x}\left(\frac{1}{x}-\frac{1}{t}\right)f'\left(t\right)dt+\frac{1}{x},x>0\]
then f (x) is equal to
a) \[e^{1/x-1}\]
b) \[e^{x-1}\]
c) \[x e^{x-1}\]
d) \[x e^{1/x-1}\]
Explanation:
9. The equation of the tangent to the curve \[y=\int_{x}^{x^{2}}\] log t dt at x=2 is
a) y – 6 log 2 = 7 log 2(x – 2)
b) \[y-\log 2e^{1/3}=\left(\log 2\right)x\]
c) \[y-8\log 2e^{-1/3}=5 \log 2x\]
d) y + 9 log 2 + 2 = (7 log 2)x.
Explanation:
10. The value of\[\int_{-1}^{1}x \mid x \mid\] dx is
a) 2
b) 1
c) 0
d) none of these
Explanation:
