Integral Calculus – I One Mark Questions

12th Standard EM

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Business Maths

Time : 00:40:00 Hrs
Total Marks : 20
    15 x 1 = 15
  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. \(\frac { { e }^{ x } }{ \sqrt { { 1+e }^{ x } } } \)dx is

    (a)

    \(\frac { { e }^{ x } }{ \sqrt { { 1+e }^{ x } } } +c\)

    (b)

    \(2\sqrt { 1+{ e }^{ x } } +c\)

    (c)

    \(\sqrt { 1+{ e }^{ x } } +c\)

    (d)

    \({ e }^{ x }\sqrt { 1+{ e }^{ x } } +c\)

  3. \(\frac { { e }^{ x } }{ { e }^{ x }+1 } \) dx

    (a)

    log\(\left| \frac { { e }^{ x } }{ { e }^{ x }+1 } \right| +c\)

    (b)

    log\(\left| \frac { { e }^{ x }+1 }{ { e }^{ x } } \right| +c\)

    (c)

    log\(\left| { e }^{ x } \right| +c\)

    (d)

    log\(\left| { e }^{ x }+1 \right| +c\)

  4. \(\int _{ 2 }^{ 4 }{ \frac { dx }{ x } } \) is

    (a)

    log 4

    (b)

    0

    (c)

    log 2

    (d)

    log 8

  5. The value of \(\int _{ \frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ cosx } \) dx is

    (a)

    0

    (b)

    2

    (c)

    1

    (d)

    4

  6. If \(\int _{ 0 }^{ 1 }{ f(x) } dx=1,\int _{ 0 }^{ 1 }{ xf(x) } dx=a\) and \(\int _{ 0 }^{ 1 }{ { x }^{ 2 }f(x) } dx={ a }^{ 2 }\), then \(\int _{ 0 }^{ 1 }{ { (a-x) }^{ 2 } } f(x)\) is

    (a)

    4a2

    (b)

    0

    (c)

    2a2

    (d)

    1

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

    (a)

    12

    (b)

    4

    (c)

    4!

    (d)

    64

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

    (a)

    -xex

    (b)

    xex

    (c)

    -xe-x

    (d)

    xe-x

  9. 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)

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

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

    (a)

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

    (b)

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

    (c)

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

    (d)

    none of these

  11. \(\int { \frac { 2 }{ { \left( { e }^{ x }+{ e }^{ -x } \right) }^{ 2 } } } \) 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)

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

  12. ∫ e3 log x (x4 +1)-1 dx = ____________ +c

    (a)

    \(\log { \left| { x }^{ 4 }+1 \right| } \)

    (b)

    \(4\log { \left| { x }^{ 4 }+1 \right| } \)

    (c)

    -4 log |x4 +1|

    (d)

    \(\frac { 1 }{ 4 } \log { \left| { x }^{ 4 }+1 \right| } \)

  13. \(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ sinx } dx=\)

    (a)

    1

    (b)

    2

    (c)

    -1

    (d)

    -2

  14. \(\int _{ 0 }^{ 1 }{ \frac { 1 }{ 2x-3 } } \) dx = ____________

    (a)

    \(\frac { 1 }{ 2 } log3\)

    (b)

    \(-\frac { 1 }{ 2 } log3\)

    (c)

    \(\frac { 1 }{ 2 } log1\)

    (d)

    0

  15. \(\int _{ 1 }^{ 4 }{ f\left( x \right) } dx,\) where f(x) =  \(\begin{cases} 7x+3\quad if\quad 1\le x\le 3 \\ 8x\quad if\quad 3\le x\le 4 \end{cases}\) is _____________.

    (a)

    58

    (b)

    60

    (c)

    62

    (d)

    52

  16. 5 x 1 = 5
  17. ∫ e-t dt

  18. (1)

    proper definite integer

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

  20. (2)

    Improper definite intgral

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

  22. (3)

    \(\Gamma (n)\quad \)

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

  24. (4)

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

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

  26. (5)

    Indefinite inegral

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