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#### Applications of Integration Model Question Paper 1

12th Standard EM

Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. If $f(x)=\int _{ 0 }^{ x }{ t\ cos\ t\ dt,\ then\ \frac { dx }{ dx } }$

(a)

cos x-x sin x

(b)

sin x+x cos x

(c)

x cos x

(d)

x sin x

2. The value of $\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 99 }dx }$ is

(a)

$\frac{1}{11000}$

(b)

$\frac{1}{10100}$

(c)

$\frac{1}{10010}$

(d)

$\frac{1}{10001}$

3. The value of  $\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx }$ is

(a)

$\frac{3\pi}{10}$

(b)

$\frac{3\pi}{8}$

(c)

$\frac{3\pi}{4}$

(d)

$\frac{3\pi}{2}$

4. The value of $\int _{ \frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \sqrt { \frac { 1-cos2x }{ 2x } } }$ dx is

(a)

$\frac { 1 }{ 2 }$

(b)

2

(c)

0

(d)

1

5. The area enclosed by the curve y = $\frac { { x }^{ 2 } }{ 2 }$ , the x - axis and the lines x = 1, x = 3 is

(a)

4

(b)

8$\frac23$

(c)

13

(d)

4$\frac{1}{3}$

6. 5 x 2 = 10
7. Evaluate: $\int ^{log 2}_{-log 2} e ^{-|x|}$ dx.

8. Evaluate the following integrals using properties of integration:
$\int _{ 0 }^{ 1 }{ |5x-3|dx }$

9. Evaluate: $\int _{ 0 }^{ 2\pi }{ { x }^{ 2 }sin\ nx\ dx }$ where n is a positive integer.

10. Evaluate $\int _{ 0 }^{ 1 }{ \frac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx }$

11. Find the volume of the solid obtained by revolving the area of the triangle whose sides are x = 4, y = 0 and 3x - 4y = 0 about x - axis

12. 5 x 3 = 15
13. Evaluate: $\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx$

14. Evaluate :$\int _{ 0 }^{ 1 }{ \frac { 2x+7 }{ { 5x }^{ 2 }+9 } } dx$

15. Evaluate $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { cot \ x } }{ \sqrt { cot \ x } +\sqrt { tan \ x } } dx }$

16. Evaluate $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sin \ x }{ 9+{ cos }^{ 2 } } dx }$

17. Evaluate $\int _{ 0 }^{ 1 }{ { xe }^{ -2x } } dx$

18. 4 x 5 = 20
19. Prove that $\int^{\frac{\pi}{4}}_{0} \frac {sin 2x dx}{ sin ^4x +cos ^4 x}$ = $\frac{\pi}{4}$

20. Evaluate : $\int ^\frac{\pi}{4}_{0} \frac{1}{sin x+cos x}$ dx

21. Find the area bounded by x = at2, y = at between the ordinates corresponding to t = 1 and t = 2

22. Using integration, find the area of the triangle with sides y = 2x+1, y = 3x + 1 and x = 4.