Mathematics - Integral Calculus

Question - 1

The integral \(\int { \frac { { 2 }^{ x+1 }-{ 5 }^{ x-1 } }{ { 10 }^{ x } } } dx\), equals

  • A \(-\frac { 2 }{ log5 } ({ 5 }^{ -x })+\frac { 1 }{ 5log2 } ({ 2 }^{ -x })+c\)
  • B \(\frac { 2 }{ log5 } ({ 5 }^{ -x })+\frac { 1 }{ 5log2 } ({ 2 }^{ -x })+c\)
  • C \(-\frac { 2 }{ log5 } ({ 5 }^{ -x })-\frac { 1 }{ 5log2 } ({ 2 }^{ -x })+c\)
  • D NONE OF THESE

Question - 2

\(\int { { (tanx+cotx) }^{ 2 } } dx,\)equals

  • A tanx+cotx+c
  • B tanx-cotx+c
  • C log(tanx+cotx)+c
  • D NONE OF THESE

Question - 3

\(\int { \frac { dx }{ \sqrt { x+1 } +\sqrt { x-1 } } } \); equals

  • A \(\frac { 1 }{ 3 } [{ (x+1) }^{ 3/2 }+{ (x-1) }^{ 3/2 }]+c\)
  • B \(\frac { 1 }{ 3 } [{ (x+1) }^{ 3/2 }-{ (x-1) }^{ 3/2 }]+c\)
  • C \(\frac { 2 }{ 3 } [{ (x-1) }^{ 3/2 }+{ (x+1) }^{ 3/2 }]+c\)
  • D \(\frac { 1}{ 3 } [{ (x-1) }^{ 3/2 }-{ (x+1) }^{ 3/2 }]\)

Question - 4

\(\int { cos } \left( \frac { x }{ 2 } \right) \sqrt { 1+cosxdx, } \) is equal to

  • A \(\frac { 1 }{ \sqrt { 2 } } (x-sinx)+c\)
  • B \(\frac { 1 }{ \sqrt { 2 } } (x+cosx)+c\)
  • C \(\frac { 1 }{ \sqrt { 2 } } (x-cosx)+c\)
  • D \(\frac { 1 }{ \sqrt { 2 } } (x+sinx)+c\)

Question - 5

The integral

\(\int { \frac { \sqrt { x } +\sqrt [ 3 ]{ x } }{ \sqrt [ 4 ]{ { x }^{ 5 } } -\sqrt [ 6 ]{ { x }^{ 7 } } } dx, } \) equals

  • A \(4{ x }^{ 1/4 }+{ 6x }^{ 1/6 }+{ 24x }^{ 1/12 }+24log\left| { x }^{ 1/12 }-1 \right| +c\)
  • B \(4{ x }^{ 1/4 }-{ 6x }^{ 1/6 }+{ 24x }^{ 1/12 }-24log\left| { x }^{ 1/12 }-1 \right| +c\)
  • C \(-4{ x }^{ 1/4 }+{ 6x }^{ 1/6 }+{ 24x }^{ 1/12 }+24log\left| { x }^{ 1/12 }-1 \right| +c\)
  • D NONE OF THESE

Question - 6

The value of the integral \(\int _{ 0 }^{ 100\pi }{ \sqrt { 1+cos\quad 2x\quad dx, } } \) is

  • A \(100\sqrt { 2 } \)
  • B \(200\sqrt { 2 } \)
  • C \(50\sqrt { 2 } \)
  • D none of these

Question - 7

The value of the integral 

\(\int _{ 0 }^{ 200 }{ (x-[x])dx } \), where [ ] denotes the greastest integer function, is

  • A 100
  • B 200
  • C 50
  • D 25

Question - 8

The value of the integral \(\int _{ 0 }^{ \pi /4 }{ \frac { \sin { \theta } +\cos { \theta } }{ 9+16\sin { 2\theta } } } d\theta \), is 

  • A \(\frac { 1 }{ 20 } log\quad 2\)
  • B log 2
  • C log 3
  • D \(\frac { 1 }{ 20 } log\quad 3\)

Question - 9

Let \(f:R\rightarrow R\) be a continuous function.Then the value of the integral 

\(\int _{ -\pi /2 }^{ \pi /2 }{ \{ f(x)+f(-x)\} \{ g(x)-g(-x)\} dx } \), is

  • A \(\pi \)
  • B 1
  • C -1
  • D 0

Question - 10

The value of integral \(\int _{ 0 }^{ \pi }{ \frac { dx }{ 1+sinx } } \) equals

  • A 0
  • B \(\frac { 1 }{ 2 } \)
  • C 1
  • D \(\frac { 3 }{ 2 } \)
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