Definite Integration · Mathematics · JEE Advanced
Numerical
1
The value of $2 \int\limits_0^{\frac{\pi}{2}} f(x) g(x) d x-\int\limits_0^{\frac{\pi}{2}} g(x) d x$ is ____________.
JEE Advanced 2024 Paper 2 Online
2
The value of $\frac{16}{\pi^3} \int\limits_0^{\frac{\pi}{2}} f(x) g(x) d x$ is ______.
JEE Advanced 2024 Paper 2 Online
3
For $x \in \mathbb{R}$, let $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then the minimum value of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\int\limits_0^{x \tan ^{-1} x} \frac{e^{(t-\cos t)}}{1+t^{2023}} d t$ is :
JEE Advanced 2023 Paper 2 Online
4
The greatest integer less than or equal to
$$ \int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}} d x $$
is ___________.
$$ \int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}} d x $$
is ___________.
JEE Advanced 2022 Paper 2 Online
5
Let $${g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2$$, and $$f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R$$ be functions such that $${g_1}(x) = 1,{g_2}(x) = |4x - \pi |$$ and $$f(x) = {\sin ^2}x$$, for all $$x \in \left[ {{\pi \over 8},{{3\pi } \over 8}} \right]$$. Define $${S_i} = \int\limits_{{\pi \over 8}}^{{{3\pi } \over 8}} {f(x).{g_i}(x)dx} $$, i = 1, 2
The value of $${{16{S_1}} \over \pi }$$ is _____________.
The value of $${{16{S_1}} \over \pi }$$ is _____________.
JEE Advanced 2021 Paper 2 Online
6
Let $${g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2$$, and $$f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R$$ be functions such that $${g_1}(x) = 1,{g_2}(x) = |4x - \pi |$$ and $$f(x) = {\sin ^2}x$$, for all $$x \in \left[ {{\pi \over 8},{{3\pi } \over 8}} \right]$$. Define $${S_i} = \int\limits_{{\pi \over 8}}^{{{3\pi } \over 8}} {f(x).{g_i}(x)dx} $$, i = 1, 2
The value of $${{48{S_2}} \over {{\pi ^2}}}$$ is ___________.
The value of $${{48{S_2}} \over {{\pi ^2}}}$$ is ___________.
JEE Advanced 2021 Paper 2 Online
7
For any real number x, let [ x ] denote the largest integer less than or equal to x. If $$I = \int\limits_0^{10} {\left[ {\sqrt {{{10x} \over {x + 1}}} } \right]dx} $$, then the value of 9I is __________.
JEE Advanced 2021 Paper 2 Online
8
Let $$f:R \to R$$ be a differentiable function such that its derivative f' is continuous and f($$\pi $$) = $$-$$6.
If $$F:[0,\pi ] \to R$$ is defined by $$F(x) = \int_0^x {f(t)dt} $$, and if $$\int_0^\pi {(f'(x)} + F(x))\cos x\,dx$$ = 2
then the value of f(0) is ...........
If $$F:[0,\pi ] \to R$$ is defined by $$F(x) = \int_0^x {f(t)dt} $$, and if $$\int_0^\pi {(f'(x)} + F(x))\cos x\,dx$$ = 2
then the value of f(0) is ...........
JEE Advanced 2020 Paper 2 Offline
9
The value of the integral $$ \int\limits_0^{\pi /2} {{{3\sqrt {\cos \theta } } \over {{{(\sqrt {\cos \theta } + \sqrt {\sin \theta } )}^5}}}} d\theta $$ equals ..............
JEE Advanced 2019 Paper 2 Offline
10
If $$I = {2 \over \pi }\int\limits_{ - \pi /4}^{\pi /4} {{{dx} \over {(1 + {e^{\sin x}})(2 - \cos 2x)}}} $$, then 27I2 equals .................
JEE Advanced 2019 Paper 1 Offline
11
The value of the integral
$$\int_0^{1/2} {{{1 + \sqrt 3 } \over {{{({{(x + 1)}^2}{{(1 - x)}^6})}^{1/4}}}}dx} $$ is ........
$$\int_0^{1/2} {{{1 + \sqrt 3 } \over {{{({{(x + 1)}^2}{{(1 - x)}^6})}^{1/4}}}}dx} $$ is ........
JEE Advanced 2018 Paper 2 Offline
12
The total number of distinct $$x \in \left[ {0,1} \right]$$ for which
$$\int\limits_0^x {{{{t^2}} \over {1 + {t^4}}}} dt = 2x - 1$$
$$\int\limits_0^x {{{{t^2}} \over {1 + {t^4}}}} dt = 2x - 1$$
JEE Advanced 2016 Paper 1 Offline
13
If $$\alpha = \int\limits_0^1 {\left( {{e^{9x + 3{{\tan }^{ - 1}}x}}} \right)\left( {{{12 + 9{x^2}} \over {1 + {x^2}}}} \right)} dx$$ where $${\tan ^{ - 1}}x$$ takes only principal values, then the value of $$\left( {{{\log }_e}\left| {1 + \alpha } \right| - {{3\pi } \over 4}} \right)$$ is
JEE Advanced 2015 Paper 2 Offline
14
Let $$f:R \to R$$ be a function defined by
$$f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2} \cr {0,} & {x > 2} \cr } } \right.$$ where $$\left[ x \right]$$ is the greatest integer less than or equal to $$x$$, if $$I = \int\limits_{ - 1}^2 {{{xf\left( {{x^2}} \right)} \over {2 + f\left( {x + 1} \right)}}dx,} $$ then the value of $$(4I-1)$$ is
$$f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2} \cr {0,} & {x > 2} \cr } } \right.$$ where $$\left[ x \right]$$ is the greatest integer less than or equal to $$x$$, if $$I = \int\limits_{ - 1}^2 {{{xf\left( {{x^2}} \right)} \over {2 + f\left( {x + 1} \right)}}dx,} $$ then the value of $$(4I-1)$$ is
JEE Advanced 2015 Paper 1 Offline
15
The value of $$\int\limits_0^1 {4{x^3}\left\{ {{{{d^2}} \over {d{x^2}}}{{\left( {1 - {x^2}} \right)}^5}} \right\}dx} $$ is
JEE Advanced 2014 Paper 1 Offline
16
For any real number $$x,$$ let $$\left[ x \right]$$ denote the largest integer less than or equal to $$x.$$ Let $$f$$ be a real valued function defined on the interval $$\left[ { - 10,10} \right]$$ by
$$$f\left( x \right) = \left\{ {\matrix{
{x - \left[ x \right]} & {if\left[ x \right]is\,odd,} \cr
{1 + \left[ x \right] - x} & {if\left[ x \right]is\,even} \cr
} } \right.$$$
Then the value of $${{{\pi ^2}} \over {10}}\int\limits_{ - 10}^{10} {f\left( x \right)\cos \,\pi x\,dx} $$ is
IIT-JEE 2010 Paper 1 Offline
17
Let $$f:R \to R$$ be a continuous function which satisfies $$f(x) = \int\limits_0^x {f(t)dt} $$. Then, the value of $$f(\ln 5)$$ is ____________.
IIT-JEE 2009 Paper 2 Offline
MCQ (Single Correct Answer)
1
Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=\sqrt{n}$ if $x \in\left[\frac{1}{n+1}, \frac{1}{n}\right)$ where $n \in \mathbb{N}$. Let $g:(0,1) \rightarrow \mathbb{R}$ be a function such that $\int\limits_{x^2}^x \sqrt{\frac{1-t}{t}} d t < g(x) < 2 \sqrt{x}$ for all $x \in(0,1)$.
Then $\lim\limits_{x \rightarrow 0} f(x) g(x)$
JEE Advanced 2023 Paper 1 Online
2
Which of the following statements is TRUE?
JEE Advanced 2021 Paper 2 Online
3
Which of the following statements is TRUE?
JEE Advanced 2021 Paper 2 Online
4
The value of $$\int\limits_{-{\pi \over 2}}^{{\pi \over 2}} {{{{x^2}\cos x} \over {1 + {e^x}}}dx} $$ is equal to
JEE Advanced 2016 Paper 2 Offline
5
Let $$f'\left( x \right) = {{192{x^3}} \over {2 + {{\sin }^4}\,\pi x}}$$ for all $$x \in R\,\,$$ with $$f\left( {{1 \over 2}} \right) = 0$$.
If $$m \le \int\limits_{1/2}^1 {f\left( x \right)dx \le M,} $$ then the possible values of $$m$$ and $$M$$ are
If $$m \le \int\limits_{1/2}^1 {f\left( x \right)dx \le M,} $$ then the possible values of $$m$$ and $$M$$ are
JEE Advanced 2015 Paper 2 Offline
6
List - $$I$$
P.$$\,\,\,\,$$ The number of polynomials $$f(x)$$ with non-negative integer coefficients of degree $$ \le 2$$, satisfying $$f(0)=0$$ and $$\int_0^1 {f\left( x \right)dx = 1,} $$ is
Q.$$\,\,\,\,$$ The number of points in the interval $$\left[ { - \sqrt {13} ,\sqrt {13} } \right]$$
at which $$f\left( x \right) = \sin \left( {{x^2}} \right) + \cos \left( {{x^2}} \right)$$ attains its maximum value, is
R.$$\,\,\,\,$$ $$\int\limits_{ - 2}^2 {{{3{x^2}} \over {\left( {1 + {e^x}} \right)}}dx} $$ equals
S.$$\,\,\,\,$$ $${{\left( {\int\limits_{ - {1 \over 2}}^{{1 \over 2}} {\cos 2x\log \left( {{{1 + x} \over {1 - x}}} \right)dx} } \right)} \over {\left( {\int\limits_0^{{1 \over 2}} {\cos 2x\log \left( {{{1 + x} \over {1 - x}}} \right)dx} } \right)}}$$
P.$$\,\,\,\,$$ The number of polynomials $$f(x)$$ with non-negative integer coefficients of degree $$ \le 2$$, satisfying $$f(0)=0$$ and $$\int_0^1 {f\left( x \right)dx = 1,} $$ is
Q.$$\,\,\,\,$$ The number of points in the interval $$\left[ { - \sqrt {13} ,\sqrt {13} } \right]$$
at which $$f\left( x \right) = \sin \left( {{x^2}} \right) + \cos \left( {{x^2}} \right)$$ attains its maximum value, is
R.$$\,\,\,\,$$ $$\int\limits_{ - 2}^2 {{{3{x^2}} \over {\left( {1 + {e^x}} \right)}}dx} $$ equals
S.$$\,\,\,\,$$ $${{\left( {\int\limits_{ - {1 \over 2}}^{{1 \over 2}} {\cos 2x\log \left( {{{1 + x} \over {1 - x}}} \right)dx} } \right)} \over {\left( {\int\limits_0^{{1 \over 2}} {\cos 2x\log \left( {{{1 + x} \over {1 - x}}} \right)dx} } \right)}}$$
List $$II$$
1.$$\,\,\,\,$$ $$8$$
2.$$\,\,\,\,$$ $$2$$
3.$$\,\,\,\,$$ $$4$$
4.$$\,\,\,\,$$ $$0$$
JEE Advanced 2014 Paper 2 Offline
7
Given that for each $$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt} $$ exists. Let this limit be $$g(a).$$ In addition, it is given that the function $$g(a)$$ is differentiable on $$(0,1).$$
The value of $$g\left( {{1 \over 2}} \right)$$ is
JEE Advanced 2014 Paper 2 Offline
8
Given that for each $$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt} $$ exists. Let this limit be $$g(a).$$ In addition, it is given that the function $$g(a)$$ is differentiable on $$(0,1).$$
The value of $$g'\left( {{1 \over 2}} \right)$$ is
JEE Advanced 2014 Paper 2 Offline
9
The following integral $$\int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\left( {2\cos ec\,\,x} \right)}^{17}}dx} $$ is equal to
JEE Advanced 2014 Paper 2 Offline
10
Let $$f$$ $$:\,\,\left[ {{1 \over 2},1} \right] \to R$$ (the set of all real number) be a positive,
non-constant and differentiable function such that
$$f'\left( x \right) < 2f\left( x \right)$$ and $$f\left( {{1 \over 2}} \right) = 1.$$ Then the value of $$\int\limits_{1/2}^1 {f\left( x \right)} \,dx$$ lies in the interval
non-constant and differentiable function such that
$$f'\left( x \right) < 2f\left( x \right)$$ and $$f\left( {{1 \over 2}} \right) = 1.$$ Then the value of $$\int\limits_{1/2}^1 {f\left( x \right)} \,dx$$ lies in the interval
JEE Advanced 2013 Paper 1 Offline
11
The value of the integral $$\int\limits_{ - \pi /2}^{\pi /2} {\left( {{x^2} + 1n{{\pi + x} \over {\pi - x}}} \right)\cos xdx} $$ is
IIT-JEE 2012 Paper 2 Offline
12
The value of $$\,\int\limits_{\sqrt {\ell n2} }^{\sqrt {\ell n3} } {{{x\sin {x^2}} \over {\sin {x^2} + \sin \left( {\ell n6 - {x^2}} \right)}}\,dx} $$ is
IIT-JEE 2011 Paper 1 Offline
13
The value of $$\int\limits_0^1 {{{{x^4}{{\left( {1 - x} \right)}^4}} \over {1 + {x^2}}}dx} $$ is (are)
IIT-JEE 2010 Paper 1 Offline
14
The value of $$\mathop {\lim }\limits_{x \to 0} {1 \over {{x^3}}}\int\limits_0^x {{{t\ln \left( {1 + t} \right)} \over {{t^4} + 4}}} dt$$ is
IIT-JEE 2010 Paper 1 Offline
15
Let $$f$$ be a real-valued function defined on the interval $$(-1, 1)$$ such that
$${e^{ - x}}f\left( x \right) = 2 + \int\limits_0^x {\sqrt {{t^4} + 1} \,\,dt,} $$ for all $$x \in \left( { - 1,1} \right)$$,
and let $${f^{ - 1}}$$ be the inverse function of $$f$$. Then $$\left( {{f^{ - 1}}} \right)'\left( 2 \right)$$ is equal to
$${e^{ - x}}f\left( x \right) = 2 + \int\limits_0^x {\sqrt {{t^4} + 1} \,\,dt,} $$ for all $$x \in \left( { - 1,1} \right)$$,
and let $${f^{ - 1}}$$ be the inverse function of $$f$$. Then $$\left( {{f^{ - 1}}} \right)'\left( 2 \right)$$ is equal to
IIT-JEE 2010 Paper 2 Offline
16
Let $$g\left( x \right) = \int\limits_0^{{e^x}} {{{f'\left( t \right)} \over {1 + {t^2}}}} \,dt.$$
Which of the following is true?
IIT-JEE 2008 Paper 2 Offline
17
$$\int\limits_{ - 1}^1 {g'\left( x \right)dx = } $$
IIT-JEE 2008 Paper 1 Offline
18
Let the definite integral be defined by the formula
$$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \right)} \right).} $$ For more accurate result for
$$c \in \left( {a,b} \right),$$ we can use $$\int\limits_a^b {f\left( x \right)dx = \int\limits_a^c {f\left( x \right)dx + \int\limits_c^b {f\left( x \right)dx = F\left( c \right)} } } $$ so
that for $$c = {{a + b} \over 2},$$ we get $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 4}\left( {f\left( a \right) + f\left( b \right) + 2f\left( c \right)} \right).} $$
$$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \right)} \right).} $$ For more accurate result for
$$c \in \left( {a,b} \right),$$ we can use $$\int\limits_a^b {f\left( x \right)dx = \int\limits_a^c {f\left( x \right)dx + \int\limits_c^b {f\left( x \right)dx = F\left( c \right)} } } $$ so
that for $$c = {{a + b} \over 2},$$ we get $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 4}\left( {f\left( a \right) + f\left( b \right) + 2f\left( c \right)} \right).} $$
If $$f''\left( x \right) < 0\,\forall x \in \left( {a,b} \right)$$ and $$c$$ is a point such that $$a < c < b,$$ and
$$\left( {c,f\left( c \right)} \right)$$ is the point lying on the curve for which $$F(c)$$ is
maximum, then $$f'(c)$$ is equal to
IIT-JEE 2006
19
Let the definite integral be defined by the formula
$$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \right)} \right).} $$ For more accurate result for
$$c \in \left( {a,b} \right),$$ we can use $$\int\limits_a^b {f\left( x \right)dx = \int\limits_a^c {f\left( x \right)dx + \int\limits_c^b {f\left( x \right)dx = F\left( c \right)} } } $$ so
that for $$c = {{a + b} \over 2},$$ we get $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 4}\left( {f\left( a \right) + f\left( b \right) + 2f\left( c \right)} \right).} $$
$$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \right)} \right).} $$ For more accurate result for
$$c \in \left( {a,b} \right),$$ we can use $$\int\limits_a^b {f\left( x \right)dx = \int\limits_a^c {f\left( x \right)dx + \int\limits_c^b {f\left( x \right)dx = F\left( c \right)} } } $$ so
that for $$c = {{a + b} \over 2},$$ we get $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 4}\left( {f\left( a \right) + f\left( b \right) + 2f\left( c \right)} \right).} $$
If $$\mathop {\lim }\limits_{x \to a} {{\int\limits_a^x {f\left( x \right)dx - \left( {{{x - a} \over 2}} \right)\left( {f\left( x \right) + f\left( a \right)} \right)} } \over {{{\left( {x - a} \right)}^3}}} = 0,\,\,$$ then $$f(x)$$ is
of maximum degree
IIT-JEE 2006
20
Let the definite integral be defined by the formula
$$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \right)} \right).} $$ For more accurate result for
$$c \in \left( {a,b} \right),$$ we can use $$\int\limits_a^b {f\left( x \right)dx = \int\limits_a^c {f\left( x \right)dx + \int\limits_c^b {f\left( x \right)dx = F\left( c \right)} } } $$ so
that for $$c = {{a + b} \over 2},$$ we get $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 4}\left( {f\left( a \right) + f\left( b \right) + 2f\left( c \right)} \right).} $$
$$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \right)} \right).} $$ For more accurate result for
$$c \in \left( {a,b} \right),$$ we can use $$\int\limits_a^b {f\left( x \right)dx = \int\limits_a^c {f\left( x \right)dx + \int\limits_c^b {f\left( x \right)dx = F\left( c \right)} } } $$ so
that for $$c = {{a + b} \over 2},$$ we get $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 4}\left( {f\left( a \right) + f\left( b \right) + 2f\left( c \right)} \right).} $$
$$\int\limits_0^{\pi /2} {\sin x\,dx = } $$
IIT-JEE 2006
21
$$\int\limits_{ - 2}^0 {\left\{ {{x^3} + 3{x^2} + 3x + 3 + \left( {x + 1} \right)\cos \left( {x + 1} \right)} \right\}\,\,dx} $$ is equal to
IIT-JEE 2005 Screening
22
The value of the integral $$\int\limits_0^1 {\sqrt {{{1 - x} \over {1 + x}}} dx} $$ is
IIT-JEE 2004 Screening
23
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
IIT-JEE 2004 Screening
24
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
IIT-JEE 2003 Screening
25
If $$f\left( x \right) = \int\limits_{{x^2}}^{{x^2} + 1} {{e^{ - {t^2}}}} dt,$$ then $$f(x)$$ increases in
IIT-JEE 2003 Screening
26
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
IIT-JEE 2002 Screening
27
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
IIT-JEE 2002 Screening
28
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
IIT-JEE 2002 Screening
29
The value of $$\int\limits_{ - \pi }^\pi {{{{{\cos }^2}x} \over {1 + {a^x}}}dx,\,a > 0,} $$ is
IIT-JEE 2001 Screening
30
If $$f\left( x \right) = \left\{ {\matrix{
{{e^{\cos x}}\sin x,} & {for\,\,\left| x \right| \le 2} \cr
{2,} & {otherwise,} \cr
} } \right.$$ then $$\int\limits_{ - 2}^3 {f\left( x \right)dx = } $$
IIT-JEE 2000 Screening
31
The value of the integral $$\int\limits_{{e^{ - 1}}}^{{e^2}} {\left| {{{{{\log }_e}x} \over x}} \right|dx} $$ is :
IIT-JEE 2000 Screening
32
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
IIT-JEE 2000 Screening
33
$$\int\limits_{\pi /4}^{3\pi /4} {{{dx} \over {1 + \cos x}}} $$ is equal to
IIT-JEE 1999
34
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
IIT-JEE 1999
35
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
IIT-JEE 1998
36
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
IIT-JEE 1998
37
The value of $$\int\limits_\pi ^{2\pi } {\left[ {2\,\sin x} \right]\,dx} $$ where [ . ] represents the greatest integer function is
IIT-JEE 1995 Screening
38
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
IIT-JEE 1995 Screening
39
The value of $$\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}} $$ is
IIT-JEE 1993
40
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
$$\int\limits_{ - \pi /2}^{\pi /2} {\left[ {f\left( x \right) + f\left( { - x} \right)} \right]\left[ {g\left( x \right) - g\left( { - x} \right)} \right]dx} $$ is
IIT-JEE 1990
41
For any integer $$n$$ the integral ...........
$$\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right)xdx} $$ has the value
$$\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right)xdx} $$ has the value
IIT-JEE 1985
42
The value of the integral $$\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx} $$ is
IIT-JEE 1983
43
The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$$
IIT-JEE 1981
44
Let $$a, b, c$$ be non-zero real numbers such that
$$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $$
Then the quadratic equation $$a{x^2} + bx + c = 0$$ has
$$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} } $$
Then the quadratic equation $$a{x^2} + bx + c = 0$$ has
IIT-JEE 1981
MCQ (More than One Correct Answer)
1
Consider the equation
$$ \int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty) $$
Which of the following statements is/are TRUE?
JEE Advanced 2022 Paper 1 Online
2
Let $$f:\left[ { - {\pi \over 2},{\pi \over 2}} \right] \to R$$ be a continuous function such that $$f(0) = 1$$ and $$\int_0^{{\pi \over 3}} {f(t)dt = 0} $$. Then which of the following statements is(are) TRUE?
JEE Advanced 2021 Paper 2 Online
3
Let b be a nonzero real number. Suppose f : R $$ \to $$ R is a differentiable function such that f(0) = 1. If the derivative f' of f satisfies the equation $$f'(x) = {{f(x)} \over {{b^2} + {x^2}}}$$
for all x$$ \in $$R, then which of the following statements is/are TRUE?
for all x$$ \in $$R, then which of the following statements is/are TRUE?
JEE Advanced 2020 Paper 2 Offline
4
Which of the following inequalities is/are TRUE?
JEE Advanced 2020 Paper 1 Offline
5
If $$I = \sum\nolimits_{k = 1}^{98} {\int_k^{k + 1} {{{k + 1} \over {x(x + 1)}}} dx} $$, then
JEE Advanced 2017 Paper 2 Offline
6
Let
$$f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \over 2}} \right)...\left( {x + {n \over n}} \right)} \over {n!\left( {{x^2} + {n^2}} \right)\left( {{x^2} + {{{n^2}} \over 4}} \right)....\left( {{x^2} + {{{n^2}} \over {{n^2}}}} \right)}}} \right)^{{x \over n}}},$$ for
all $$x>0.$$ Then
$$f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \over 2}} \right)...\left( {x + {n \over n}} \right)} \over {n!\left( {{x^2} + {n^2}} \right)\left( {{x^2} + {{{n^2}} \over 4}} \right)....\left( {{x^2} + {{{n^2}} \over {{n^2}}}} \right)}}} \right)^{{x \over n}}},$$ for
all $$x>0.$$ Then
JEE Advanced 2016 Paper 2 Offline
7
Let $$f\left( x \right) = 7{\tan ^8}x + 7{\tan ^6}x - 3{\tan ^4}x - 3{\tan ^2}x$$ for all $$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right).$$
Then the correct expression(s) is (are)
Then the correct expression(s) is (are)
JEE Advanced 2015 Paper 2 Offline
8
The option(s) with the values of a and $$L$$ that satisfy the following equation is (are)
$$${{\int\limits_0^{4\pi } {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} } \over {\int\limits_0^\pi {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} }} = L?$$$
JEE Advanced 2015 Paper 2 Offline
9
Let $$f:\left( {0,\infty } \right) \to R$$ be given by $$f\left( x \right) $$= $$\int\limits_{{1 \over x}}^x {{{{e^{ - \left( {t + {1 \over t}} \right)}}} \over t}} dt$$. Then
JEE Advanced 2014 Paper 1 Offline
10
Let a $$\in$$ R and f : R $$\to$$ R be given by f(x) = x5 $$-$$ 5x + a. Then,
JEE Advanced 2014 Paper 1 Offline
11
If $${I_n} = \int\limits_{ - \pi }^\pi {{{\sin nx} \over {(1 + {\pi ^x})\sin x}}dx,n = 0,1,2,} $$ .... then
IIT-JEE 2009 Paper 2 Offline
Subjective
1
Match the integrals in Column $$I$$ with the values in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS$$.
Column $$I$$
(A) $$\int\limits_{ - 1}^1 {{{dx} \over {1 + {x^2}}}} $$
(B) $$\int\limits_0^1 {{{dx} \over {\sqrt {1 - {x^2}} }}} $$
(C) $$\int\limits_2^3 {{{dx} \over {1 - {x^2}}}} $$
(D) $$\int\limits_1^2 {{{dx} \over {x\sqrt {{x^2} - 1} }}} $$
Column $$II$$
(p) $${1 \over 2}\log \left( {{2 \over 3}} \right)$$
(q) $$2\log \left( {{2 \over 3}} \right)$$
(r) $${{\pi \over 3}}$$
(s) $${{\pi \over 2}}$$
IIT-JEE 2007
2
The value of $$5050{{\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx} \over {\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx}}$$ is.
IIT-JEE 2006
3
Evaluate $$\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos \left( {{1 \over 2}\cos x} \right)} \right)\sin x\,\,dx$$
IIT-JEE 2005
4
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 $$x = \pi $$
IIT-JEE 2004
5
Find the value of $$\int\limits_{ - \pi /3}^{\pi /3} {{{\pi + 4{x^3}} \over {2 - \cos \left( {\left| x \right| + {\pi \over 3}} \right)}}dx} $$
IIT-JEE 2004
6
If $$f$$ is an even function then prove that
$$\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\sin 2x} \right)\cos x\,dx.} $$
$$\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\sin 2x} \right)\cos x\,dx.} $$
IIT-JEE 2003
7
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, $$\ln t = {\log _e}t$$.
IIT-JEE 2000
8
Integrate $$\int\limits_0^\pi {{{{e^{\cos x}}} \over {{e^{\cos x}} + {e^{ - \cos x}}}}\,dx.} $$
IIT-JEE 1999
9
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
$$\int_0^1 {{{\tan }^{ - 1}}\left( {1 - x + {x^2}} \right)dx.} $$
IIT-JEE 1998
10
Determine the value of $$\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.$$
IIT-JEE 1997
11
Evaluate the definite integral :
$$$\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \right){{\cos }^{ - 1}}\left( {{{2x} \over {1 + {x^2}}}} \right)} dx$$$
IIT-JEE 1995
12
Let $${I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.$$ Use mathematical induction to prove that $${I_m} = m\,\pi ,m = 0,1,2,........$$
IIT-JEE 1995
13
Show that $$\int\limits_0^{n\pi + v} {\left| {\sin x} \right|dx = 2n + 1 - \cos \,v} $$ where $$n$$ is a positive integer and $$\,0 \le v < \pi .$$
IIT-JEE 1994
14
Evaluate $$\int_2^3 {{{2{x^5} + {x^4} - 2{x^3} + 2{x^2} + 1} \over {\left( {{x^2} + 1} \right)\left( {{x^4} - 1} \right)}}} dx.$$
IIT-JEE 1993
15
Determine a positive integer $$n \le 5,$$ such that
$$$\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e} $$$
IIT-JEE 1992
16
Evaluate $$\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx} $$
IIT-JEE 1991
17
Show that $$\int\limits_0^{\pi /2} {f\left( {\sin 2x} \right)\sin x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\cos 2x} \right)\cos x\,dx} $$
IIT-JEE 1990
18
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 $$\int\limits_0^{\pi /2} {\sin 2kx\,\cot \,x\,dx = {\pi \over 2}} $$
$${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \right]$$
Hence prove that $$\int\limits_0^{\pi /2} {\sin 2kx\,\cot \,x\,dx = {\pi \over 2}} $$
IIT-JEE 1990
19
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 $$\int\limits_0^a {f\left( x \right)g\left( x \right)dx = \int\limits_0^a {f\left( x \right)dx} } $$
$$f\left( x \right) = f\left( {a - x} \right)$$ and $$g\left( x \right) + g\left( {a - x} \right) = 2,$$
then show that $$\int\limits_0^a {f\left( x \right)g\left( x \right)dx = \int\limits_0^a {f\left( x \right)dx} } $$
IIT-JEE 1989
20
Evaluate $$\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx} $$
IIT-JEE 1988
21
Evaluate : $$\int\limits_0^\pi {{{x\,dx} \over {1 + \cos \,\alpha \,\sin x}},0 < \alpha < \pi } $$
IIT-JEE 1986
22
Evaluate the following : $$\,\,\int\limits_0^{\pi /2} {{{x\sin x\cos x} \over {{{\cos }^4}x + {{\sin }^4}x}}} dx$$
IIT-JEE 1985
23
Given a function $$f(x)$$ such that
(i) it is integrable over every interval on the real line and
(ii) $$f(t+x)=f(x),$$ for every $$x$$ and a real $$t$$, then show that
the integral $$\int\limits_a^{a + 1} {f\,\,\left( x \right)} \,dx$$ is independent of a.
(i) it is integrable over every interval on the real line and
(ii) $$f(t+x)=f(x),$$ for every $$x$$ and a real $$t$$, then show that
the integral $$\int\limits_a^{a + 1} {f\,\,\left( x \right)} \,dx$$ is independent of a.
IIT-JEE 1984
24
Evaluate the following $$\int\limits_0^{{1 \over 2}} {{{x{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx} $$
IIT-JEE 1984
25
Evaluate : $$\int\limits_0^{\pi /4} {{{\sin x + \cos x} \over {9 + 16\sin 2x}}dx} $$
IIT-JEE 1983
26
Show that $$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} = {\pi \over 2}\int\limits_0^\pi {f\left( {\sin x} \right)dx.} $$
IIT-JEE 1982
27
Find the value of $$\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx} $$
IIT-JEE 1982
28
Show that : $$\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n}}} \right) = \log 6$$
IIT-JEE 1981
Fill in the Blanks
1
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 ............
IIT-JEE 1997
2
The value of $$\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx} $$ is ...............
IIT-JEE 1997
3
For $$n>0,$$ $$\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx = $$
IIT-JEE 1996
4
If for nonzero $$x$$, $$af(x)+$$ $$bf\left( {{1 \over x}} \right) = {1 \over x} - 5$$ where $$a \ne b,$$ then
$$\int_1^2 {f\left( x \right)dx} = .......$$
$$\int_1^2 {f\left( x \right)dx} = .......$$
IIT-JEE 1996
5
The value of $$\int\limits_2^3 {{{\sqrt x } \over {\sqrt {3 - x} + \sqrt x }}} dx$$ is ...........
IIT-JEE 1994
6
The value of $$\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi } $$ is ..............
IIT-JEE 1993
7
The value of $$\int\limits_{ - 2}^2 {\left| {1 - {x^2}} \right|dx} $$ is ...............
IIT-JEE 1989
8
The integral $$\int\limits_0^{1.5} {\left[ {{x^2}} \right]dx,} $$
Where [ ] denotes the greatest integer function, equals .............
IIT-JEE 1988
9
$$f\left( x \right) = \left| {\matrix{
{\sec x} & {\cos x} & {{{\sec }^2}x + \cot x\cos ec\,x} \cr
{{{\cos }^2}x} & {{{\cos }^2}x} & {\cos e{c^2}x} \cr
1 & {{{\cos }^2}x} & {{{\cos }^2}x} \cr
} } \right|.$$
Then $$\int\limits_0^{\pi /2} {f\left( x \right)dx = .......} $$
Then $$\int\limits_0^{\pi /2} {f\left( x \right)dx = .......} $$
IIT-JEE 1987