Integral Calculus – I Important Questions

12th Standard EM

    Reg.No. :
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Business Maths

Time : 01:00:00 Hrs
Total Marks : 50
    10 x 1 = 10
  1. \(\frac { 1 }{ { x }^{ 3 } } \)dx is

    (a)

    \(\frac { -3 }{ { x }^{ 2 } } +c\)

    (b)

    \(\frac { -1 }{ 2{ x }^{ 2 } } +c\)

    (c)

    \(\frac { -1 }{ { 3x }^{ 2 } } +c\)

    (d)

    \(\frac { -2 }{ { x }^{ 2 } } +c\)

  2. ഽ2xdx is

    (a)

    2x log 2 + c

    (b)

    2x + c

    (c)

    \(\frac { 2^{ x } }{ log2 } +c\)

    (d)

    \(\frac { log2 }{ { 2 }^{ x } } +c\)

  3. \(\frac { sin2x }{ 2sinx } dx\) is

    (a)

    sin x + c

    (b)

    \(\frac12\)sin x + c

    (c)

    cos x + c

    (d)

    \(\frac12\)cos x + c

  4. \(\Gamma \left( \frac { 3 }{ 2 } \right) \)

    (a)

    \(\sqrt { \pi } \)

    (b)

    \(\frac { \sqrt { \pi } }{ 2 } \)

    (c)

    \(2\sqrt { \pi } \)

    (d)

    \(\frac32\)

  5. \(\int _{ 0 }^{ \infty }{ { x }^{ 4 }{ e }^{ -x } } \)dx is

    (a)

    12

    (b)

    4

    (c)

    4!

    (d)

    64

  6. \(\int { \left( x-1 \right) } { e }^{ -x }\) dx = __________ +c

    (a)

    -xex

    (b)

    xex

    (c)

    -xe-x

    (d)

    xe-x

  7. If \(\int { \frac { { 2 }^{ \frac { 1 }{ x } } }{ { x }^{ 2 } } } dx=k{ 2 }^{ \frac { 1 }{ x } }\) +c, then k is

    (a)

    \(-\frac { 1 }{ { log }_{ e }2 } \)

    (b)

    - loge2

    (c)

    -1

    (d)

    \(\frac { 1 }{ 2 } \)

  8. \(\int { { \left| x \right| }^{ 3 } } \)dx = ________________ +c

    (a)

    \(\frac { { -x }^{ 4 } }{ 4 } \)

    (b)

    \(\frac { { \left| x \right| }^{ 4 } }{ 4 } \)

    (c)

    \(\frac { { x }^{ 4 } }{ 4 } \)

    (d)

    none of these

  9. \(\int { \frac { 2 }{ { \left( { e }^{ x }+{ e }^{ -x } \right) }^{ 2 } } } \) dx = ____________ +c 

    (a)

    \(\frac { { -e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } \)

    (b)

    \(-\frac { { -e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } \)

    (c)

    \(\frac { 1 }{ { \left( { e }^{ x }+1 \right) }^{ 2 } } \)

    (d)

    \(\frac { 1 }{ { e }^{ x }-{ e }^{ -x } } \)

  10. \(\int { { e }^{ x } } \) (1-cot x +cot2 x) dx = _______________ +c

    (a)

    ex cot x

    (b)

    - ex cot x

    (c)

    ex cosec x

    (d)

    -ex cosec x

  11. 5 x 1 = 5
  12. ∫ e-t dt

  13. (1)

    proper definite integer

  14. \(\int _{ 0 }^{ 1 }{ { e }^{ -t } } dt\quad \)

  15. (2)

    \(\Gamma (n)\quad \)

  16. \(\int _{ 0 }^{ \infty }{ { e }^{ -t } } dt\quad \)

  17. (3)

    \(\frac { n! }{ { a }^{ n+1 } } \)

  18. For n > 0, \(\int _{ 0 }^{ \infty }{ { x }^{ n-1 }{ e }^{ -x } } \) dx

  19. (4)

    Improper definite intgral

  20. \(\int _{ 0 }^{ \infty }{ { x }^{ n }{ e }^{ -ax } } \) dx where n is a positive integer

  21. (5)

    Indefinite inegral

    5 x 2 = 10
  22. Integrate the following with respect to x.
    \(\sqrt { 3x+5 } \)

  23.  Integrate the following with respect to x.

    \({ \left( { 9x }^{ 2 }-\frac { 4 }{ { x }^{ 2 } } \right) }^{ 2 }\)

  24.  Integrate the following with respect to x.

    \(\sqrt{x}\)(x3 − 2x + 3)

  25. Evaluate \(\int { { a }^{ 3{ log }_{ a }x } } dx\)

  26. Evaluate \(\int { \frac { 2{ cos }^{ 2 }x-cos2x }{ { sin }^{ 2 }x } } \)

  27. 5 x 3 = 15
  28.  Evaluate \(\int { \frac { { ax }^{ 2 }+bx+v }{ \sqrt { x } } dx } \)

  29. Evaluate \(\int { \sqrt { 2x+1dx } } \)

  30. Evaluate \(\int { \frac { dx }{ { \left( 2x+3 \right) }^{ 2 } } } \)

  31. Evaluate  \(\int { \frac { { { x }^{ 4 }+{ x }^{ 4 }+1 } }{ { x }^{ 2 }-x+1 } } \)

  32. Evaluate \(\int { \frac { cos2x-cos2\alpha }{ cosx-cos\alpha } } dx\)

  33. 2 x 5 = 10
  34. Evaluate \(\int { \frac { { 3x }^{ 2 }+2x+1 }{ x } dx } \)

  35. If f'(x) = a sin x + b cos x and f'(0) = 4, f(0) = 3, f\(\left( \frac { \pi }{ 2 } \right) \) = 5, find f(x).

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