Definite Integration
579 Questions
2005
JEE Mains
MCQ
AIEEE 2005
The value of $\int\limits_{ - \pi }^\pi {{{{{\cos }^2}} \over {1 + {a^x}}}dx,\,\,a > 0,} $ is
A.
$a\,\pi $
B.
${\pi \over 2}$
C.
${\pi \over a}$
D.
${2\pi }$
2005
JEE Advanced
Numerical
IIT-JEE 2005
Evaluate $\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos \left( {{1 \over 2}\cos x} \right)} \right)\sin x\,\,dx$
Correct Answer: $${{24} \over 5}\left[ {e\cos \left( {{1 \over 2}} \right) + {1 \over 2}e\sin \left( {{1 \over 2}} \right) - 1} \right]$$
2005
JEE Advanced
MCQ
IIT-JEE 2005 Screening
$\int\limits_{ - 2}^0 {\left\{ {{x^3} + 3{x^2} + 3x + 3 + \left( {x + 1} \right)\cos \left( {x + 1} \right)} \right\}\,\,dx} $ is equal to
A.
$-4$
B.
$0$
C.
$4$
D.
$6$
2005
JEE Advanced
MCQ
IIT-JEE 2005 Mains
Evatuate:
$\int_\limits{0}^{\pi} e^{|\cos x|}\left[2 \sin \left(\frac{1}{2} \cos x\right)+3 \cos \left(\frac{1}{2} \cos x\right)\right] \sin x ~d x$
A.
${e \over 5}\left[ {\cos \left( {{1 \over 2}} \right) + \left( {{1 \over 2}} \right)\sin \left( {{1 \over 2}} \right) - 1} \right]$
B.
$24{e \over 5}\left[ {\cos \left( {{1 \over 2}} \right) + \left( {{1 \over 2}} \right)\sin \left( {{1 \over 2}} \right) - 1} \right]$
C.
$12{e \over 5}\left[ {\cos \left( {{1 \over 2}} \right) + \left( {{1 \over 2}} \right)\sin \left( {{1 \over 2}} \right) - 1} \right]$
D.
$5{e \over 5}\left[ {\cos \left( {{1 \over 2}} \right) + \left( {{1 \over 2}} \right)\sin \left( {{1 \over 2}} \right) - 1} \right]$
2004
JEE Mains
MCQ
AIEEE 2004
$\mathop {Lim}\limits_{n \to \infty } \sum\limits_{r = 1}^n {{1 \over n}{e^{{r \over n}}}} $ is
A.
$e+1$
B.
$e-1$
C.
$1-e$
D.
$e$
2004
JEE Mains
MCQ
AIEEE 2004
If $\int\limits_0^\pi {xf\left( {\sin x} \right)dx = A\int\limits_0^{\pi /2} {f\left( {\sin x} \right)dx,} } $ then $A$ is
A.
$2\pi $
B.
$\pi $
C.
${\pi \over 4}$
D.
$0$
2004
JEE Mains
MCQ
AIEEE 2004
The value of $I = \int\limits_0^{\pi /2} {{{{{\left( {\sin x + \cos x} \right)}^2}} \over {\sqrt {1 + \sin 2x} }}dx} $ is
A.
$3$
B.
$1$
C.
$2$
D.
$0$
2004
JEE Mains
MCQ
AIEEE 2004
The value of $\int\limits_{ - 2}^3 {\left| {1 - {x^2}} \right|dx} $ is
A.
${1 \over 3}$
B.
${14 \over 3}$
C.
${7 \over 3}$
D.
${28 \over 3}$
2004
JEE Mains
MCQ
AIEEE 2004
If $f\left( x \right) = {{{e^x}} \over {1 + {e^x}}},{I_1} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {xg\left\{ {x\left( {1 - x} \right)} \right\}dx} $
and ${I_2} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {g\left\{ {x\left( {1 - x} \right)} \right\}dx} ,$ then the value of ${{{I_2}} \over {{I_1}}}$ is
and ${I_2} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {g\left\{ {x\left( {1 - x} \right)} \right\}dx} ,$ then the value of ${{{I_2}} \over {{I_1}}}$ is
A.
$1$
B.
$-3$
C.
$-1$
D.
$2$
2004
JEE Advanced
Numerical
IIT-JEE 2004
If $y\left( x \right) = \int\limits_{{x^2}/16}^{{x^2}} {{{\cos x\cos \sqrt \theta } \over {1 + {{\sin }^2}\sqrt \theta }}d\theta ,} $ then find ${{dy} \over {dx}}$ at $x = \pi $
Correct Answer: $$2\pi $$
2004
JEE Advanced
Numerical
IIT-JEE 2004
Find the value of $\int\limits_{ - \pi /3}^{\pi /3} {{{\pi + 4{x^3}} \over {2 - \cos \left( {\left| x \right| + {\pi \over 3}} \right)}}dx} $
Correct Answer: $${{4\pi } \over {\sqrt 3 }}\left[ {{{\tan }^{ - 1}}3 - {\pi \over 4}} \right]$$
2004
JEE Advanced
MCQ
IIT-JEE 2004 Screening
The value of the integral $\int\limits_0^1 {\sqrt {{{1 - x} \over {1 + x}}} dx} $ is
A.
${\pi \over 2} + 1$
B.
${\pi \over 2} - 1$
C.
$-1$
D.
$1$
2004
JEE Advanced
MCQ
IIT-JEE 2004 Screening
If $f(x)$ is differentiable and $\int\limits_0^{{t^2}} {xf\left( x \right)dx = {2 \over 5}{t^5},} $ then $f\left( {{4 \over {25}}} \right)$ equals
A.
$2/5$
B.
$-5/2$
C.
$1$
D.
$5/2$
2003
JEE Mains
MCQ
AIEEE 2003
If $f\left( y \right) = {e^y},$ $g\left( y \right) = y;y > 0$ and
$F\left( t \right) = \int\limits_0^t {f\left( {t - y} \right)g\left( y \right)dy,} $ then :
$F\left( t \right) = \int\limits_0^t {f\left( {t - y} \right)g\left( y \right)dy,} $ then :
A.
$F\left( t \right) = t{e^{ - t}}$
B.
$F\left( t \right) = 1t - t{e^{ - 1}}\left( {1 + t} \right)$
C.
$F\left( t \right) = {e^t} - \left( {1 + t} \right)$
D.
$F\left( t \right) = t{e^t}$.
2003
JEE Mains
MCQ
AIEEE 2003
Let $f(x)$ be a function satisfying $f'(x)=f(x)$ with $f(0)=1$ and $g(x)$ be a function that satisfies $f\left( x \right) + g\left( x \right) = {x^2}$. Then the value of the integral $\int\limits_0^1 {f\left( x \right)g\left( x \right)dx,} $ is
A.
$e + {{{e^2}} \over 2} + {5 \over 2}$
B.
$e - {{{e^2}} \over 2} - {5 \over 2}$
C.
$e + {{{e^2}} \over 2} - {3 \over 2}$
D.
$e - {{{e^2}} \over 2} - {3 \over 2}$
2003
JEE Mains
MCQ
AIEEE 2003
The value of the integral $I = \int\limits_0^1 {x{{\left( {1 - x} \right)}^n}dx} $ is
A.
${1 \over {n + 1}} + {1 \over {n + 2}}$
B.
${1 \over {n + 1}}$
C.
${1 \over {n + 2}}$
D.
${1 \over {n + 1}} - {1 \over {n + 2}}$
2003
JEE Mains
MCQ
AIEEE 2003
$\mathop {\lim }\limits_{n \to \infty } {{1 + {2^4} + {3^4} + .... + {n^4}} \over {{n^5}}}$ - $\mathop {\lim }\limits_{n \to \infty } {{1 + {2^3} + {3^3} + .... + {n^3}} \over {{n^5}}}$
A.
${1 \over 5}$
B.
${1 \over 30}$
C.
zero
D.
${1 \over 4}$
2003
JEE Mains
MCQ
AIEEE 2003
If $f\left( {a + b - x} \right) = f\left( x \right)$ then $\int\limits_a^b {xf\left( x \right)dx} $ is equal to
A.
${{a + b} \over 2}\int\limits_a^b {f\left( {a + b + x} \right)dx} $
B.
${{a + b} \over 2}\int\limits_a^b {f\left( {b - x} \right)dx} $
C.
${{a + b} \over 2}\int\limits_a^b {f\left( x \right)dx} $
D.
$\,{{b - a} \over 2}\int\limits_a^b {f\left( x \right)dx} $
2003
JEE Mains
MCQ
AIEEE 2003
The value of $\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {{{\sec }^2}tdt} } \over xsinx}$ is
A.
0
B.
3
C.
2
D.
1
2003
JEE Advanced
Numerical
IIT-JEE 2003
If $f$ is an even function then prove that
$\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\sin 2x} \right)\cos x\,dx.} $
$\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\sin 2x} \right)\cos x\,dx.} $
Correct Answer: Solve it.
2003
JEE Advanced
MCQ
IIT-JEE 2003 Screening
If $l\left( {m,n} \right) = \int\limits_0^1 {{t^m}{{\left( {1 + t} \right)}^n}dt,} $ then the expression for $l(m, n)$ in terms of $l(m+n, n-1)$ is
A.
${{{2^n}} \over {m + 1}} - {n \over {m + 1}}l\left( {m + 1,n - 1} \right)$
B.
${n \over {m + 1}}l\left( {m + 1,n - 1} \right)$
C.
${{{2^n}} \over {m + 1}} + {n \over {m + 1}}l\left( {m + 1,n - 1} \right)$
D.
${m \over {n + 1}}l\left( {m + 1,n - 1} \right)$
2003
JEE Advanced
MCQ
IIT-JEE 2003 Screening
If $f\left( x \right) = \int\limits_{{x^2}}^{{x^2} + 1} {{e^{ - {t^2}}}} dt,$ then $f(x)$ increases in
A.
$(-2, 2)$
B.
no value of $x$
C.
$\left( {0,\infty } \right)$
D.
$\left( { - \infty ,0} \right)$
2002
JEE Mains
MCQ
AIEEE 2002
If $y=f(x)$ makes +$ve$ intercept of $2$ and $0$ unit on $x$ and $y$ axes and encloses an area of $3/4$ square unit with the axes then $\int\limits_0^2 {xf'\left( x \right)dx} $ is
A.
$3/2$
B.
$1$
C.
$5/4$
D.
$-3/4$
2002
JEE Mains
MCQ
AIEEE 2002
$\int\limits_0^2 {\left[ {{x^2}} \right]dx} $ is
A.
$2 - \sqrt 2 $
B.
$2 + \sqrt 2 $
C.
$\,\sqrt 2 - 1$
D.
$ - \sqrt 2 - \sqrt 3 + 5$
2002
JEE Mains
MCQ
AIEEE 2002
${I_n} = \int\limits_0^{\pi /4} {{{\tan }^n}x\,dx} $ then $\,\mathop {\lim }\limits_{n \to \infty } \,n\left[ {{I_n} + {I_{n + 2}}} \right]$ equals
A.
${1 \over 2}$
B.
$1$
C.
$\infty $
D.
zero
2002
JEE Mains
MCQ
AIEEE 2002
$\int\limits_0^{10\pi } {\left| {\sin x} \right|dx} $ is
A.
$20$
B.
$8$
C.
$10$
D.
$18$
2002
JEE Mains
MCQ
AIEEE 2002
$\int_{ - \pi }^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} dx$ is
A.
${{{\pi ^2}} \over 4}$
B.
${{\pi ^2}}$
C.
zero
D.
${\pi \over 2}$
2002
JEE Mains
MCQ
AIEEE 2002
$\mathop {\lim }\limits_{n \to \infty } {{{1^p} + {2^p} + {3^p} + ..... + {n^p}} \over {{n^{p + 1}}}}$ is
A.
${1 \over {p + 1}}$
B.
${1 \over {1 - p}}$
C.
${1 \over p} - {1 \over {p - 1}}$
D.
${1 \over {p + 2}}$
2002
JEE Advanced
MCQ
IIT-JEE 2002 Screening
The integral $\int\limits_{ - 1/2}^{1/2} {\left( {\left[ x \right] + \ell n\left( {{{1 + x} \over {1 - x}}} \right)} \right)dx} $ equal to
A.
$ - {1 \over 2}$
B.
$0$
C.
$1$
D.
$2\ell n\left( {{1 \over 2}} \right)$
2002
JEE Advanced
MCQ
IIT-JEE 2002 Screening
Let $T>0$ be a fixed real number . Suppose $f$ is a continuous
function such that for all $x \in R$, $f\left( {x + T} \right) = f\left( x \right)$.
function such that for all $x \in R$, $f\left( {x + T} \right) = f\left( x \right)$.
If $I = \int\limits_0^T {f\left( x \right)dx} $ then the value of $\int\limits_3^{3 + 3T} {f\left( {2x} \right)dx} $ is
A.
$3/2I$
B.
$2I$
C.
$3I$
D.
$6I$
2002
JEE Advanced
MCQ
IIT-JEE 2002 Screening
Let $T>0$ be a fixed real number . Suppose $f$ is a continuous
function such that for all $x \in R$, $f\left( {x + T} \right) = f\left( x \right)$.
function such that for all $x \in R$, $f\left( {x + T} \right) = f\left( x \right)$.
If $I = \int\limits_0^T {f\left( x \right)dx} $ then the value of $\int\limits_3^{3 + 3T} {f\left( {2x} \right)dx} $ is
A.
$3/2I$
B.
$2I$
C.
$3I$
D.
$6I$
2001
JEE Advanced
MCQ
IIT-JEE 2001 Screening
The value of $\int\limits_{ - \pi }^\pi {{{{{\cos }^2}x} \over {1 + {a^x}}}dx,\,a > 0,} $ is
A.
$\pi $
B.
$a\pi $
C.
$\pi /2$
D.
$2\pi $
2000
JEE Advanced
Numerical
IIT-JEE 2000
For $x>0,$ let $f\left( x \right) = \int\limits_e^x {{{\ln t} \over {1 + t}}dt.} $ Find the function
$f\left( x \right) + f\left( {{1 \over x}} \right)$ and show that $f\left( e \right) + f\left( {{1 \over e}} \right) = {1 \over 2}.$
Here, $\ln t = {\log _e}t$.
$f\left( x \right) + f\left( {{1 \over x}} \right)$ and show that $f\left( e \right) + f\left( {{1 \over e}} \right) = {1 \over 2}.$
Here, $\ln t = {\log _e}t$.
Correct Answer: Solve it.
2000
JEE Advanced
MCQ
IIT-JEE 2000 Screening
If $f\left( x \right) = \left\{ {\matrix{
{{e^{\cos x}}\sin x,} & {for\,\,\left| x \right| \le 2} \cr
{2,} & {otherwise,} \cr
} } \right.$ then $\int\limits_{ - 2}^3 {f\left( x \right)dx = } $
A.
$0$
B.
$1$
C.
$2$
D.
$3$
2000
JEE Advanced
MCQ
IIT-JEE 2000 Screening
The value of the integral $\int\limits_{{e^{ - 1}}}^{{e^2}} {\left| {{{{{\log }_e}x} \over x}} \right|dx} $ is :
A.
$3/2$
B.
$5/2$
C.
$3$
D.
$5$
2000
JEE Advanced
MCQ
IIT-JEE 2000 Screening
Let $g\left( x \right) = \int\limits_0^x {f\left( t \right)dt,} $ where f is such that
${1 \over 2} \le f\left( t \right) \le 1,$ for $t \in \left[ {0,1} \right]$ and $\,0 \le f\left( t \right) \le {1 \over 2},$ for $t \in \left[ {1,2} \right]$.
Then $g(2)$ satisfies the inequality
${1 \over 2} \le f\left( t \right) \le 1,$ for $t \in \left[ {0,1} \right]$ and $\,0 \le f\left( t \right) \le {1 \over 2},$ for $t \in \left[ {1,2} \right]$.
Then $g(2)$ satisfies the inequality
A.
$ - {3 \over 2} \le g\left( 2 \right) < {1 \over 2}$
B.
$0 \le g\left( 2 \right) < 2$
C.
${3 \over 2} < g\left( 2 \right) \le {5 \over 2}$
D.
$2 < g\left( 2 \right) < 4$
1999
JEE Advanced
Numerical
IIT-JEE 1999
Integrate $\int\limits_0^\pi {{{{e^{\cos x}}} \over {{e^{\cos x}} + {e^{ - \cos x}}}}\,dx.} $
Correct Answer: $${\pi \over 2}$$
1999
JEE Advanced
MCQ
IIT-JEE 1999
If for a real number $y$, $\left[ y \right]$ is the greatest integer less than or
equal to $y$, then the value of the integral $\int\limits_{\pi /2}^{3\pi /2} {\left[ {2\sin x} \right]dx} $ is
equal to $y$, then the value of the integral $\int\limits_{\pi /2}^{3\pi /2} {\left[ {2\sin x} \right]dx} $ is
A.
$ - \pi $
B.
$0$
C.
$ - \pi /2$
D.
$ \pi /2$
1999
JEE Advanced
MCQ
IIT-JEE 1999
$\int\limits_{\pi /4}^{3\pi /4} {{{dx} \over {1 + \cos x}}} $ is equal to
A.
$2$
B.
$-2$
C.
$1/2$
D.
$-1/2$
1998
JEE Advanced
Numerical
IIT-JEE 1998
Prove that $\int_0^1 {{{\tan }^{ - 1}}} \,\left( {{1 \over {1 - x + {x^2}}}} \right)dx = 2\int_0^1 {{{\tan }^{ - 1}}} \,x\,dx.$
Hence or otherwise, evaluate the integral
$\int_0^1 {{{\tan }^{ - 1}}\left( {1 - x + {x^2}} \right)dx.} $
Hence or otherwise, evaluate the integral
$\int_0^1 {{{\tan }^{ - 1}}\left( {1 - x + {x^2}} \right)dx.} $
Correct Answer: $$\log 2$$
1998
JEE Advanced
MCQ
IIT-JEE 1998
Let $f\left( x \right) = x - \left[ x \right],$ for every real number $x$, where $\left[ x \right]$ is the integral part of $x$. Then $\int_{ - 1}^1 {f\left( x \right)\,dx} $ is
A.
$1$
B.
$2$
C.
$0$
D.
$1/2$
1998
JEE Advanced
MCQ
IIT-JEE 1998
If $\int_0^x {f\left( t \right)dt = x + \int_x^1 {t\,\,f\left( t \right)\,\,dt,} } $ then the value of $f(1)$ is
A.
$1/2$
B.
$0$
C.
$1$
D.
$-1/2$
1997
JEE Advanced
Numerical
IIT-JEE 1997
Determine the value of $\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.$
Correct Answer: $${\pi ^2}$$
1997
JEE Advanced
Numerical
IIT-JEE 1997
The value of $\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx} $ is ...............
Correct Answer: $$2$$
1997
JEE Advanced
Numerical
IIT-JEE 1997
Let ${d \over {dx}}\,F\left( x \right) = {{{e^{\sin x}}} \over x},\,x > 0.$ If $\int_1^4 {{{2{e^{\sin {x^2}}}} \over x}} \,\,dx = F\left( k \right) - F\left( 1 \right)$
then one of the possible values of $k$ is ............
then one of the possible values of $k$ is ............
Correct Answer: $$16$$
1996
JEE Advanced
Numerical
IIT-JEE 1996
For $n>0,$ $\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx = $
Correct Answer: $${\pi ^2}$$
1996
JEE Advanced
Numerical
IIT-JEE 1996
If for nonzero $x$, $af(x)+$ $bf\left( {{1 \over x}} \right) = {1 \over x} - 5$ where $a \ne b,$ then
$\int_1^2 {f\left( x \right)dx} = .......$
$\int_1^2 {f\left( x \right)dx} = .......$
Correct Answer: $${1 \over {{a^2} - {b^2}}}\left[ {a\left( {\log 2 - 5} \right) + {{7b} \over 2}} \right]$$
1995
JEE Advanced
Numerical
IIT-JEE 1995
Let ${I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.$ Use mathematical induction to prove that ${I_m} = m\,\pi ,m = 0,1,2,........$
Correct Answer: Solve it.
1995
JEE Advanced
Numerical
IIT-JEE 1995
Evaluate the definite integral :
$$\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \right){{\cos }^{ - 1}}\left( {{{2x} \over {1 + {x^2}}}} \right)} dx$$
Correct Answer: $${\pi \over {12}}\left[ {\pi + 3{{\log }_e}\left( {2 + \sqrt 3 } \right) - 4\sqrt 3 } \right]$$
1995
JEE Advanced
MCQ
IIT-JEE 1995 Screening
If $f\left( x \right)\,\,\, = \,\,\,A\sin \left( {{{\pi x} \over 2}} \right)\,\,\, + \,\,\,B,\,\,\,f'\left( {{1 \over 2}} \right) = \sqrt 2 $ and
$\int\limits_0^1 {f\left( x \right)dx = {{2A} \over \pi },} $ then constants $A$ and $B$ are
$\int\limits_0^1 {f\left( x \right)dx = {{2A} \over \pi },} $ then constants $A$ and $B$ are
A.
${\pi \over 2}$ and ${\pi \over 2}$
B.
${2 \over \pi }$ and ${3 \over \pi }$
C.
$0$ and ${-4 \over \pi }$
D.
${4 \over \pi }$ and $0$