# Definite Integration · Mathematics · JEE Advanced

Start Practice

## Numerical

JEE Advanced 2024 Paper 2 Online
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
The value of $\frac{16}{\pi^3} \int\limits_0^{\frac{\pi}{2}} f(x) g(x) d x$ is ______.
JEE Advanced 2023 Paper 2 Online
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} \righ... JEE Advanced 2022 Paper 2 Online The greatest integer less than or equal to $$\int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}}... JEE Advanced 2021 Paper 2 Online Let$${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$$be fu... JEE Advanced 2021 Paper 2 Online Let$${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$$be fu... JEE Advanced 2021 Paper 2 Online For any real number x, let [ x ] denote the largest integer less than or equal to x. If$$I = \int\limits_0^{10} {\left[ {\sqrt {{{10x} \over {x + 1}}... JEE Advanced 2020 Paper 2 Offline 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 b... JEE Advanced 2019 Paper 2 Offline The value of the integral $$\int\limits_0^{\pi /2} {{{3\sqrt {\cos \theta } } \over {{{(\sqrt {\cos \theta } + \sqrt {\sin \theta } )}^5}}}} d\theta... JEE Advanced 2019 Paper 1 Offline If$$I = {2 \over \pi }\int\limits_{ - \pi /4}^{\pi /4} {{{dx} \over {(1 + {e^{\sin x}})(2 - \cos 2x)}}} $$, then 27I2 equals .................... JEE Advanced 2018 Paper 2 Offline The value of the integral$$\int_0^{1/2} {{{1 + \sqrt 3 } \over {{{({{(x + 1)}^2}{{(1 - x)}^6})}^{1/4}}}}dx} $$is ........ JEE Advanced 2016 Paper 1 Offline The total number of distinct$$x \in \left[ {0,1} \right]$$for which$$\int\limits_0^x {{{{t^2}} \over {1 + {t^4}}}} dt = 2x - 1$$JEE Advanced 2015 Paper 2 Offline If$$\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 ^... JEE Advanced 2015 Paper 1 Offline Let $$f:R \to R$$ be a function defined by $$f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2} \cr {0,} & {x > ... JEE Advanced 2014 Paper 1 Offline The value of$$\int\limits_0^1 {4{x^3}\left\{ {{{{d^2}} \over {d{x^2}}}{{\left( {1 - {x^2}} \right)}^5}} \right\}dx} $$is IIT-JEE 2010 Paper 1 Offline For any real number$$x,$$let$$\left[ x \right]$$denote the largest integer less than or equal to$$x.$$Let$$f$$be a real valued function define... IIT-JEE 2009 Paper 2 Offline Let$$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)$$is ____________.... ## MCQ (Single Correct Answer) JEE Advanced 2023 Paper 1 Online Let f:(0,1) \rightarrow \mathbb{R} be the function defined as f(x)=\sqrt{n} if x \in\left[\frac{1}{n+1}, \frac{1}{n}\right) where n \in \mathbb... JEE Advanced 2021 Paper 2 Online Which of the following statements is TRUE? JEE Advanced 2021 Paper 2 Online Which of the following statements is TRUE? JEE Advanced 2016 Paper 2 Offline The value of$$\int\limits_{-{\pi \over 2}}^{{\pi \over 2}} {{{{x^2}\cos x} \over {1 + {e^x}}}dx} $$is equal to JEE Advanced 2015 Paper 2 Offline Let$$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 \l... JEE Advanced 2014 Paper 2 Offline 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 List - $$I$$ P.$$\,\,\,\,$$ The number of polynomials $$f(x)$$ with non-negative integer coefficients of degree $$\le 2$$, satisfying $$f(0)=0$$ and ... JEE Advanced 2014 Paper 2 Offline Given that for each $$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \rig... JEE Advanced 2014 Paper 2 Offline Given that for each$$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \rig... JEE Advanced 2013 Paper 1 Offline 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 tha... IIT-JEE 2012 Paper 2 Offline 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 2011 Paper 1 Offline The value of $$\,\int\limits_{\sqrt {\ell n2} }^{\sqrt {\ell n3} } {{{x\sin {x^2}} \over {\sin {x^2} + \sin \left( {\ell n6 - {x^2}} \right)}}\,dx}$$... IIT-JEE 2010 Paper 1 Offline 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 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 2 Offline Let $$f$$ be a real-valued function defined on the interval $$(-1, 1)$$ such that $${e^{ - x}}f\left( x \right) = 2 + \int\limits_0^x {\sqrt {{t^4} +... IIT-JEE 2008 Paper 2 Offline Let$$g\left( x \right) = \int\limits_0^{{e^x}} {{{f'\left( t \right)} \over {1 + {t^2}}}} \,dt.$$Which of the following is true? IIT-JEE 2008 Paper 1 Offline$$\int\limits_{ - 1}^1 {g'\left( x \right)dx = } $$IIT-JEE 2006 Let the definite integral be defined by the formula$$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \... IIT-JEE 2006 Let the definite integral be defined by the formula $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \... IIT-JEE 2006 Let the definite integral be defined by the formula$$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \... 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 IIT-JEE 2004 Screening The value of the integral $$\int\limits_0^1 {\sqrt {{{1 - x} \over {1 + x}}} dx}$$ is 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 $$f\left( {{4 \over {25}}} \right)$$ equ... 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 $$l(m+n, n-1)$$... 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 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 IIT-JEE 2002 Screening Let $$T>0$$ be a fixed real number . Suppose $$f$$ is a continuous function such that for all $$x \in R$$, $$f\left( {x + T} \right) = f\left( x ... IIT-JEE 2002 Screening Let$$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 ... IIT-JEE 2001 Screening The value of $$\int\limits_{ - \pi }^\pi {{{{{\cos }^2}x} \over {1 + {a^x}}}dx,\,a > 0,}$$ is IIT-JEE 2000 Screening If $$f\left( x \right) = \left\{ {\matrix{ {{e^{\cos x}}\sin x,} & {for\,\,\left| x \right| \le 2} \cr {2,} & {otherwise,} \cr } }... IIT-JEE 2000 Screening Let$$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 \le... IIT-JEE 2000 Screening The value of the integral $$\int\limits_{{e^{ - 1}}}^{{e^2}} {\left| {{{{{\log }_e}x} \over x}} \right|dx}$$ is : 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 $$\int\limits_{\... IIT-JEE 1999$$\int\limits_{\pi /4}^{3\pi /4} {{{dx} \over {1 + \cos x}}} $$is equal to IIT-JEE 1998 If$$\int_0^x {f\left( t \right)dt = x + \int_x^1 {t\,\,f\left( t \right)\,\,dt,} } $$then the value of$$f(1)$$is IIT-JEE 1998 Let$$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_{... 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 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$...
IIT-JEE 1993
The value of $$\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}}$$ is
IIT-JEE 1990
Let $$f:R \to R$$ and $$\,\,g:R \to R$$ be continuous functions. Then the value of the integral $$\int\limits_{ - \pi /2}^{\pi /2} {\left[ {f\left(... IIT-JEE 1985 For any integer$$n$$the integral ...........$$\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right)xdx} $$has the value IIT-JEE 1983 The value of the integral$$\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx} $$is IIT-JEE 1981 The value of the definite integral$$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$$IIT-JEE 1981 Let$$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\limi...

## MCQ (More than One Correct Answer)

JEE Advanced 2022 Paper 1 Online
Consider the equation $$\int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} ... JEE Advanced 2021 Paper 2 Online Let$$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)d...
JEE Advanced 2020 Paper 2 Offline
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 equ...
JEE Advanced 2020 Paper 1 Offline
Which of the following inequalities is/are TRUE?
JEE Advanced 2017 Paper 2 Offline
If $$I = \sum\nolimits_{k = 1}^{98} {\int_k^{k + 1} {{{k + 1} \over {x(x + 1)}}} dx}$$, then
JEE Advanced 2016 Paper 2 Offline
Let $$f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \over 2}} \right)...\left( {x + ... JEE Advanced 2015 Paper 2 Offline Let$$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).$$... JEE Advanced 2015 Paper 2 Offline The option(s) with the values of a and$$L$$that satisfy the following equation is (are)$$${{\int\limits_0^{4\pi } {{e^t}\left( {{{\sin }^6}at + {{... JEE Advanced 2014 Paper 1 Offline Let $$f:\left( {0,\infty } \right) \to R$$ be given by $$f\left( x \right)$$= $$\int\limits_{{1 \over x}}^x {{{{e^{ - \left( {t + {1 \over t}} \right... JEE Advanced 2014 Paper 1 Offline Let a$$\in$$R and f : R$$\to$$R be given by f(x) = x5$$-$$5x + a. Then, IIT-JEE 2009 Paper 2 Offline If$${I_n} = \int\limits_{ - \pi }^\pi {{{\sin nx} \over {(1 + {\pi ^x})\sin x}}dx,n = 0,1,2,} $$.... then ## Subjective IIT-JEE 2007 Match the integrals in Column$$I$$with the values in Column$$II$$and indicate your answer by darkening the appropriate bubbles in the$$4 \times 4... IIT-JEE 2006 The value of $$5050{{\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx} \over {\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx... IIT-JEE 2005 Evaluate$$\,\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} \r... 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 $$... IIT-JEE 2004 Find the value of$$\int\limits_{ - \pi /3}^{\pi /3} {{{\pi + 4{x^3}} \over {2 - \cos \left( {\left| x \right| + {\pi \over 3}} \right)}}dx} $$IIT-JEE 2003 If$$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... IIT-JEE 2000 For $$x>0,$$ let $$f\left( x \right) = \int\limits_e^x {{{\ln t} \over {1 + t}}dt.}$$ Find the function $$f\left( x \right) + f\left( {{1 \over x... IIT-JEE 1999 Integrate$$\int\limits_0^\pi {{{{e^{\cos x}}} \over {{e^{\cos x}} + {e^{ - \cos x}}}}\,dx.} $$IIT-JEE 1998 Prove that$$\int_0^1 {{{\tan }^{ - 1}}} \,\left( {{1 \over {1 - x + {x^2}}}} \right)dx = 2\int_0^1 {{{\tan }^{ - 1}}} \,x\,dx.$$Hence or otherwise, ... IIT-JEE 1997 Determine the value of$$\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.$$IIT-JEE 1995 Let$${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,........ IIT-JEE 1995 Evaluate the definite integral : $$\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \right){{\cos }^{ - 1}}\left( {{{2x... IIT-JEE 1994 Show that$$\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 .$...
IIT-JEE 1993
Evaluate $$\int_2^3 {{{2{x^5} + {x^4} - 2{x^3} + 2{x^2} + 1} \over {\left( {{x^2} + 1} \right)\left( {{x^4} - 1} \right)}}} dx.$$
IIT-JEE 1992
Determine a positive integer $$n \le 5,$$ such that $$\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e}$$$IIT-JEE 1991 Evaluate $$\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx}$$ IIT-JEE 1990 Prove that for any positive integer $$k$$, $${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \rig... IIT-JEE 1990 Show that$$\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} $$IIT-JEE 1989 If$$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 \ri... IIT-JEE 1988 Evaluate $$\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx}$$ IIT-JEE 1986 Evaluate : $$\int\limits_0^\pi {{{x\,dx} \over {1 + \cos \,\alpha \,\sin x}},0 < \alpha < \pi }$$ IIT-JEE 1985 Evaluate the following : $$\,\,\int\limits_0^{\pi /2} {{{x\sin x\cos x} \over {{{\cos }^4}x + {{\sin }^4}x}}} dx$$ IIT-JEE 1984 Evaluate the following $$\int\limits_0^{{1 \over 2}} {{{x{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx}$$ IIT-JEE 1984 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$...
IIT-JEE 1983
Evaluate : $$\int\limits_0^{\pi /4} {{{\sin x + \cos x} \over {9 + 16\sin 2x}}dx}$$
IIT-JEE 1982
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
Find the value of $$\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx}$$
IIT-JEE 1981
Show that : $$\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n}}} \right) = \log 6$$

## Fill in the Blanks

IIT-JEE 1997
The value of $$\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx}$$ is ...............
IIT-JEE 1997
Let $${d \over {dx}}\,F\left( x \right) = {{{e^{\sin x}}} \over x},\,x > 0.$$ If $$\int_1^4 {{{2{e^{\sin {x^2}}}} \over x}} \,\,dx = F\left( k \rig... IIT-JEE 1996 If for nonzero$$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} = ......
IIT-JEE 1996
For $$n>0,$$ $$\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx =$$
IIT-JEE 1994
The value of $$\int\limits_2^3 {{{\sqrt x } \over {\sqrt {3 - x} + \sqrt x }}} dx$$ is ...........
IIT-JEE 1993
The value of $$\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi }$$ is ..............
IIT-JEE 1989
The value of $$\int\limits_{ - 2}^2 {\left| {1 - {x^2}} \right|dx}$$ is ...............
IIT-JEE 1988
The integral $$\int\limits_0^{1.5} {\left[ {{x^2}} \right]dx,}$$ Where [ ] denotes the greatest integer function, equals .............
IIT-JEE 1987
EXAM MAP
Medical
NEET