### 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 $\frac { 3 }{ 2 }$