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#### Applications of Integration - Two Marks Study Materials

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 30
15 x 2 = 30
1. Evaluate $\int _{ 0 }^{ x }{ { x }^{ 2 } }$cos nxdx, where n is a positive integer.

2. Evaluate the following:
$\int _{ 0 }^{ 1 }{ { x }^{ 3 }{ e }^{ -2x }dx }$

3. Evaluate $\int ^\frac {\pi}{2}_{0}$( sin2 x + cos4 x ) dx

4. Evaluate the following
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x{ cos }^{ 4 }xdx }$

5. Evaluate the following
$\int _{ 0 }^{ 1 }{ { x }^{ 2 }{ (1-x) }^{ 3 }dx }$

6. Find, by integration, the volume of the solid generated by revolving about the y-axis, the region enclosed by x2 y =1+ and y = 3.

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

8. Evaluate $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ 2x }cosxdx }$

9. Evaluate $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ -x } }$

10. Evaluate $\int _{ 0 }^{ 1 }{ \left( \cfrac { { e }^{ 5logx }-{ e }^{ 4logx } }{ { e }^{ 3logx }-{ e }^{ 2logx } } \right) }$

11. If $\int _{ 0 }^{ \infty }{ \cfrac { { x }^{ 2 }dx }{ \left( { x }^{ 2 }+{ a }^{ 2 } \right) \left( { x }^{ 2 }+{ b }^{ 2 } \right) \left( { x }^{ 2 }+{ c }^{ 2 } \right) } } =\cfrac { \pi }{ 2(a+b)(b+c)(c+a) }$ then find $\int _{ 0 }^{ \infty }{ \cfrac { dx }{ \left( { x }^{ 2 }+4 \right) \left( { x }^{ 2 }+9 \right) } }$

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

13. Find the area bounded by y=x2+2,x-x-axis, x=1 and x=2

14. Find the area enclosed between the parabola y2=4ax and the line x=a,x=9a.

15. Find the area bounded by the curve y=cosax in one arc of the curve.