iCON Education - Definite Integration

2006 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 2006

$ \text { The value of } 5050 \frac{\int_0^1\left(1-x^{50}\right)^{100} d x}{\int_0^{\frac{1}{1}}\left(1-x^{50}\right)^{101} d x} \text { is : } $

2006 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 2006

If $a_n=\frac{3}{4}-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\cdots \cdots(-1)^{n-1}\left(\frac{3}{4}\right)^n$ and $b_n=1-a_n$, then find the minimum natural number $n_0$ such that $b_n>a_n \forall n>n_0$

2005 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 2005

Evaluate $\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos \left( {{1 \over 2}\cos x} \right)} \right)\sin x\,\,dx$

2005 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 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

iCON Education HYD, 79930 92826, 73309 72826 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.

$5{e \over 5}\left[ {\cos \left( {{1 \over 2}} \right) + \left( {{1 \over 2}} \right)\sin \left( {{1 \over 2}} \right) - 1} \right]$

Correct Answer: B

Explanation:

$\begin{aligned} & \int_\limits{0}^{\pi} e^{|\cos x|}\left[2 \sin \left(\frac{1}{2} \cos x\right)+3 \cos \left(\frac{1}{2} \cos x\right)\right] \sin x d x \\ & =\int_\limits{0}^{\pi} e^{|\cos x|} 2 \sin x \sin \left(\frac{1}{2} \cos x\right) d x \\ & +\int_\limits{0}^{\pi} e^{|\cos x|} 3 \cos \left(\frac{1}{2} \cos x\right) \sin x d x \end{aligned}$

We know that

$\begin{aligned} & \int_{0}^{2 a} f(x) d x=0 \text { if } f(2 a-x)=-f(x) \\ & \quad=2 \int_{0}^{a} f(x) d x \text { if } f(2 a-x)=f(x) \\ & \quad=0+2 \int_{0}^{\frac{\pi}{2}} e^{|\cos x|} \cdot 3 \cdot \cos \left(\frac{1}{2} \cos x\right) \sin x d x \\ & \quad=6 \int_{0}^{\frac{\pi}{2}} e^{|\cos x|} \cdot \cos \left(\frac{\cos x}{2}\right) \sin x d x \end{aligned}$

Now put $\quad \cos x=t$

$-\sin x d x=d t$

when,

$\begin{aligned} & x=0 t=1 \\ & x=\frac{\pi}{2} t=0 \end{aligned}$

$\begin{aligned} & =-6 \int_\limits{1}^{0} e^{t} \cos \left(\frac{t}{2}\right) d t \\ & =6 \int_\limits{0}^{1} e^{t} \cos \left(\frac{t}{2}\right) d t \end{aligned}$

Let, $\mathrm{I}=\int e^{t} \cos \left(\frac{t}{2}\right) d t$

$=e^{t} 2 \sin \left(\frac{t}{2}\right)-2 \int e^{t} \sin \left(\frac{t}{2}\right) d t$

$ =2 e^{t} \sin \left(\frac{t}{2}\right)-2\left[e^{t}\left(\frac{-\cos \frac{t}{2}}{\frac{1}{2}}\right)+2 \int e^{t} \cos \frac{t}{2} d t\right]$

$=2 e^{t} \sin \left(\frac{t}{2}\right)+4 e^{t} \cos \left(\frac{t}{2}\right)-4 \mathrm{I}$

$5 \mathrm{I}=2 e^{t} \sin \left(\frac{t}{2}\right)+4 e^{t} \cos \left(\frac{t}{2}\right)$

$I=\frac{1}{5}\left[2 e^{t} \sin \left(\frac{t}{2}\right)+4 e^{t} \cos \left(\frac{t}{2}\right)\right]$

Required Integration

$\begin{aligned} & =\frac{6}{5}\left[2 e \sin \left(\frac{1}{2}\right)+4 e\left(\cos \frac{1}{2}\right)-4 e\right] \\ & =24 \frac{e}{5}\left[\cos \left(\frac{1}{2}\right)+\left(\frac{1}{2}\right) \sin \left(\frac{1}{2}\right)-1\right] \end{aligned}$

2004 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 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 $

2004 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 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} $

2004 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 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

iCON Education HYD, 79930 92826, 73309 72826 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 Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 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.} $

2003 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 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

iCON Education HYD, 79930 92826, 73309 72826 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 Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 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

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 2002 Screening

Let $T>0$ be a fixed real number . Suppose $f$ is a continuous
function such that for all $x \in R$, $f\left( {x + T} \right) = f\left( x \right)$.

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

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 2002 Screening

Let $T>0$ be a fixed real number . Suppose $f$ is a continuous
function such that for all $x \in R$, $f\left( {x + T} \right) = f\left( x \right)$.

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

iCON Education HYD, 79930 92826, 73309 72826 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

iCON Education HYD, 79930 92826, 73309 72826 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, $\ln t = {\log _e}t$.

2000 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 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

iCON Education HYD, 79930 92826, 73309 72826 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

iCON Education HYD, 79930 92826, 73309 72826 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

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

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1999

Integrate $\int\limits_0^\pi {{{{e^{\cos x}}} \over {{e^{\cos x}} + {e^{ - \cos x}}}}\,dx.} $

1999 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 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

A.

$ - \pi $

B.

$0$

C.

$ - \pi /2$

D.

$ \pi /2$

1999 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 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

iCON Education HYD, 79930 92826, 73309 72826 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.} $

1998 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 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

iCON Education HYD, 79930 92826, 73309 72826 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

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1997

Determine the value of $\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.$

1997 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1997

The value of $\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx} $ is ...............

1997 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 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 ............

1996 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1996

For $n>0,$ $\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx = $

1996 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1996

If for nonzero $x$, $af(x)+$ $bf\left( {{1 \over x}} \right) = {1 \over x} - 5$ where $a \ne b,$ then
$\int_1^2 {f\left( x \right)dx} = .......$

1995 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1995

Let ${I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.$ Use mathematical induction to prove that ${I_m} = m\,\pi ,m = 0,1,2,........$

1995 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1995

Evaluate the definite integral : $$\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \right){{\cos }^{ - 1}}\left( {{{2x} \over {1 + {x^2}}}} \right)} dx$$

1995 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 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

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 $0$

1995 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1995 Screening

The value of $\int\limits_\pi ^{2\pi } {\left[ {2\,\sin x} \right]\,dx} $ where [ . ] represents the greatest integer function is

A.

${{ - 5\pi } \over 3}$

B.

$\pi $

C.

${{ 5\pi } \over 3}$

D.

$ - 2\pi $

1994 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1994

Show that $\int\limits_0^{n\pi + v} {\left| {\sin x} \right|dx = 2n + 1 - \cos \,v} $ where $n$ is a positive integer and $\,0 \le v < \pi .$

1994 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1994

The value of $\int\limits_2^3 {{{\sqrt x } \over {\sqrt {3 - x} + \sqrt x }}} dx$ is ...........

1993 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1993

Evaluate $\int_2^3 {{{2{x^5} + {x^4} - 2{x^3} + 2{x^2} + 1} \over {\left( {{x^2} + 1} \right)\left( {{x^4} - 1} \right)}}} dx.$

1993 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1993

The value of $\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}} $ is

A.

$0$

B.

$1$

C.

$\pi /2$

D.

$\pi /4$

1993 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1993

The value of $\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi } $ is ..............

1992 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1992

Determine a positive integer $n \le 5,$ such that $$\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e} $$

1991 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1991

Evaluate $\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx} $

1990 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1990

Prove that for any positive integer $k$,
${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \right]$
Hence prove that $\int\limits_0^{\pi /2} {\sin 2kx\,\cot \,x\,dx = {\pi \over 2}} $

1990 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1990

Show that $\int\limits_0^{\pi /2} {f\left( {\sin 2x} \right)\sin x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\cos 2x} \right)\cos x\,dx} $

1990 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1990

Let $f:R \to R$ and $\,\,g:R \to R$ be continuous functions. Then the value of the integral
$\int\limits_{ - \pi /2}^{\pi /2} {\left[ {f\left( x \right) + f\left( { - x} \right)} \right]\left[ {g\left( x \right) - g\left( { - x} \right)} \right]dx} $ is

A.

$\pi $

B.

$1$

C.

$-1$

D.

$0$

1989 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1989

If $f$ and $g$ are continuous function on $\left[ {0,a} \right]$ satisfying
$f\left( x \right) = f\left( {a - x} \right)$ and $g\left( x \right) + g\left( {a - x} \right) = 2,$
then show that $\int\limits_0^a {f\left( x \right)g\left( x \right)dx = \int\limits_0^a {f\left( x \right)dx} } $

1989 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1989

The value of $\int\limits_{ - 2}^2 {\left| {1 - {x^2}} \right|dx} $ is ...............

1988 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1988

Evaluate $\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx} $

1988 JEE Advanced Numerical

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1988

The integral $\int\limits_0^{1.5} {\left[ {{x^2}} \right]dx,} $

Where [ ] denotes the greatest integer function, equals .............

1988 JEE Advanced MCQ

iCON Education HYD, 79930 92826, 73309 72826 IIT-JEE 1988

The value of the integral $\int\limits_0^{2a} {[{{f\left( x \right)} \over {\left\{ {f\left( x \right) + f\left( {2a - x} \right)} \right\}}}]\,dx} $ is equal to $a$.

A.

TRUE

B.

FALSE

Mathematics Chapters

Definite Integration

Explanation:

Explanation:

Explanation: