JEE Main Mathematics - Integral Calculus
Exam Duration: 60 Mins Total Questions : 30
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
\(\int { { (tanx+cotx) }^{ 2 } } dx,\)equals
- (a)
tanx+cotx+c
- (b)
tanx-cotx+c
- (c)
log(tanx+cotx)+c
- (d)
NONE OF THESE
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
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
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
The value of integral \(\int _{ 0 }^{ \pi }{ \frac { dx }{ 1+sinx } } \) equals
- (a)
0
- (b)
\(\frac { 1 }{ 2 } \)
- (c)
1
- (d)
\(\frac { 3 }{ 2 } \)
\(\int { \frac { (x+1){ (x+logx) }^{ 2 } }{ 2x } } dx,\) equals
- (a)
\(\frac { 1 }{ 6 } { (x+logx) }^{ 3 }+c\)
- (b)
\(\frac { 1 }{ 3 } { (x+logx) }^{ 3 }+c\)
- (c)
\(\frac { 1 }{ 2 } { (x+logx) }^{ 3 }+c\)
- (d)
\(\frac { 1 }{ 8 } { (x+logx) }^{ 3 }+c\)
If \(y=\sqrt { 5x-6-{ x }^{ 2 } } =1\), then \(\int _{ 2 }^{ 3 }{ y\quad dx } \) equals
- (a)
\(\frac { \pi }{ 4 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\( \pi \)
- (d)
none of these
The value of the integral \(\int _{ 0 }^{ \pi /2 }{ sin2xlogtanxdx } \) equals
- (a)
-1
- (b)
0
- (c)
1
- (d)
none of these
The integral \(\int { \frac { sin2x }{ { (a+bcosx) }^{ 2 } } dx } \) equals
- (a)
\(\frac { 1 }{ { b }^{ 2 } } [log(a+bcosx)]+c\)
- (b)
\(-\frac { 1 }{ { b }^{ 2 } } \left[ log(a+bcosx)+\frac { a }{ (a+bcosx) } \right] +c\)
- (c)
\(\frac { 1 }{ { b }^{ 2 } } \left[ log(a+bcosx)-\frac { a }{ (a+bcosx) } \right] +c\)
- (d)
\(-\frac { 2 }{ { b }^{ 2 } } \left[ log(a+bcosx)+\frac { a }{ (a+bcosx) } \right] +c\)
\(\int { \frac { sinx+cosx }{ \sqrt { 1+sin2x } } dx, } \) equals
- (a)
log(sinx+cosx)+c
- (b)
x+c
- (c)
log x+c
- (d)
\(\sqrt { 1+{ sin }^{ 2 }+c } \)
\(\begin{matrix} lim \\ x\rightarrow 0 \end{matrix}\int _{ 0 }^{ { x }^{ 2 } }{ \frac { sin\sqrt { x } }{ { x }^{ 3 } } } dx\) equals
- (a)
0
- (b)
\(\frac { 1 }{ 2 } \)
- (c)
1
- (d)
none of these
The point of maxima or minima of the function
\(f(x)=\int _{ 0 }^{ x }{ \frac { sint }{ t } } dx\) are given by
- (a)
\(n\pi \)
- (b)
\((2n+1)\frac { \pi }{ 2 } \)
- (c)
\(2n\pi \)
- (d)
none of these
If f(a+b-x)=f(x), then \(\int _{ a }^{ b }{ xf(x)dx } \) is equal to
- (a)
\(\frac { a+b }{ 2 } \int _{ a }^{ b }{ f(b-x)dx } \)
- (b)
\(\frac { a+b }{ 2 } \int _{ a }^{ b }{ f(x)dx } \)
- (c)
\(\frac { b-a }{ 2 } \int _{ a }^{ b }{ f(x)dx } \)
- (d)
none of these
\(\int _{ 0 }^{ \pi /2 }{ { sin }^{ 6 }x } { cos }^{ 4 }xdx,\) equals
- (a)
\(\frac { 3\pi }{ 256 } \)
- (b)
\(\frac { \pi }{ 128 } \)
- (c)
\(\frac { \pi }{ 64 } \)
- (d)
none of these
The value of the integral \(\int _{ -1/2 }^{ 1/2 }{ cosxln\left( \frac { 1+x }{ 1-x } \right) } dx\), equals
- (a)
0
- (b)
1
- (c)
\(\frac { 1 }{ 2 } \)
- (d)
\(-\frac { 1 }{ 2 } \)
The integral \(\int _{ 0 }^{ 1 }{ \frac { { tan }^{ -1 }x }{ x } } dx\), equals
- (a)
\(2\int _{ 0 }^{ \pi /2 }{ \frac { t }{ sint } } dt\)
- (b)
\(\int _{ 0 }^{ \pi /2 }{ \frac { t }{ sint } } dt\)
- (c)
\(\frac { 1 }{ 2 } \int _{ 0 }^{ \pi /2 }{ \frac { t }{ sint } } dt\)
- (d)
none of these
\(\int { (cos2x)log\left( \frac { cosx+sinx }{ cosx-sinx } \right) } dx,\) equals
- (a)
\(\frac { 1 }{ 2 } \left[ sin2xlog\left( \frac { cosx+sinx }{ cosx-sinx } \right) \right] \)
- (b)
\(\frac { 1 }{ 2 } log(cos2x)+c\)
- (c)
\(\frac { 1 }{ 2 } sin(2x)log\left( \frac { cosx+sinx }{ cosx-sinx } \right) \)
- (d)
NONE OF THESE
\(\int { \frac { d\theta }{ { sin }^{ 4 }\theta +{ cos }^{ 4 }\theta } } ,\)equals
- (a)
\({ tan }^{ -1 }\left( \frac { { tan }^{ 2 }\theta -1 }{ \sqrt { 2 } tan\theta } \right) +c\)
- (b)
\({ tan }^{ -1 }\left( \frac { { tan }^{ 2 }\theta -1 }{ \sqrt { 2 } } \right) +c\)
- (c)
\(\frac { 1 }{ \sqrt { 2 } } { tan }^{ -1 }\left( \frac { { tan }^{ 2 }\theta -1 }{ \sqrt { 2 } tan\theta } \right) +c\)
- (d)
NONE OF THESE
If \(\int _{ 0 }^{ 1 }{ { e }^{ { x }^{ 2 } } } (x-\alpha )dx=0\) then
- (a)
\(0<\alpha <1\)
- (b)
\(\alpha <0\)
- (c)
\(1<\alpha <2\)
- (d)
\(\alpha =0\)
\(\int _{ 2 }^{ 3 }{ \frac { \sqrt { x } }{ \sqrt { 5-x } +\sqrt { x } } } dx\),equals
- (a)
\(\frac { 1 }{ 2 } \)
- (b)
2
- (c)
3
- (d)
4
\(\begin{matrix} lim \\ n\rightarrow \infty \end{matrix}\left[ \left( 1+\frac { 1 }{ n } \right) \left( 1+\frac { 2 }{ n } \right) \left( 1+\frac { 3 }{ n } \right) ...\left( 1+\frac { n }{ n } \right) \right] ^{ 1/n }\), equals
- (a)
\(\frac { 4 }{ e } \)
- (b)
\(\frac { 3 }{ e } \)
- (c)
\(\frac { 2 }{ e } \)
- (d)
\(\frac { 1 }{ e } \)
Area commonto the parabolas
x2=4ay,y2=4ax, is
- (a)
\(\frac { 4 }{ 3 } \)a2
- (b)
\(\frac { { 8a }^{ 2 } }{ 3 } \)
- (c)
\(\frac { { 14a }^{ 2 } }{ 3 } \)
- (d)
\(\frac { { 16a }^{ 2 } }{ 3 } \)
The integral
\(-\int { \frac { { sin }^{ -1 }x-{ cos }^{ -1 }x }{ { sin }^{ -1 }x+{ cos }^{ -1 }x } dx, } \) equals
- (a)
\(\frac { 4 }{ \pi } \left( x{ sin }^{ -1 }-\sqrt { 1-{ x }^{ 2 } } \right) -x+c\)
- (b)
\(\frac { 4 }{ \pi } \left( x{ sin }^{ -1 }+\sqrt { 1-{ x }^{ 2 } } \right) -x+c\)
- (c)
\(x{ sin }^{ -1 }+\sqrt { 1-{ x }^{ 2 } } -x+c\)
- (d)
NONE OF THESE
\(\int { \left( \sqrt { tanx } +\sqrt { cotx } \right) dx, } \) equals
- (a)
\(\sqrt { 2 } { sin }^{ -1 }(sinx+cosx)+c\)
- (b)
\( { sin }^{ -1 }(sinx-cosx)+c\)
- (c)
\(\sqrt { 2 } { sin }^{ -1 }(sinx-cosx)+c\)
- (d)
NONE OF THESE
\(\int { \frac { dx }{ \sqrt { { sin }^{ 3 }xsin(x+\alpha ) } } , } \) equls
- (a)
\(-2cosec\alpha \sqrt { cos\alpha +cotxsin\alpha } +c\)
- (b)
\(-2sec\alpha \sqrt { cos\alpha +cotxsin\alpha } \)
- (c)
\(2cosec\alpha \sqrt { cos\alpha cotx +sin\alpha } \)
- (d)
NONE OF THESE
The integral \(\int { { x }^{ -2/3 }{ (1+{ x }^{ 1/2 }) }^{ -5/3 } } dx,\) equals
- (a)
\(\frac { 3 }{ { (1+{ x }^{ -1/2 }) }^{ 2/3 } } +c\)
- (b)
\(\frac { 1}{ { (1+{ x }^{ -1/2 }) }^{ 2/3 } } +c\)
- (c)
\(\frac { 2 }{ { (1+{ x }^{ -1/2 }) }^{ 2/3 } } +c\)
- (d)
NONE OF THESE
The integral
\(\int _{ -1/2 }^{ 1/2 }{ \left( [x]+ln\left( \frac { 1+x }{ 1-x } \right) \right) } dx\),equals
- (a)
\(-\frac { 1 }{ 2 } \)
- (b)
0
- (c)
1
- (d)
2ln(1/2)
The value of \(\int _{ \pi /2 }^{ 3\pi /2 }{ [2sinx]dx } \), where [ ] represents the greatest integer function, is
- (a)
\(-\pi \)
- (b)
0
- (c)
\(-\frac { \pi }{ 2 } \)
- (d)
\(\frac { \pi }{ 2 } \)