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JEE Advanced

Definite Integrals and Applications of Integrals

Mathematics

Previous Years Questions

Numerical

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For any real number x, let [ x ] denote the largest integer less than or equal to x. If $$I = \int\limits_0^{10} {\left[...
JEE Advanced 2021 Paper 2 Online
Let $${g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2$$, and $$f:\left[ {{\pi \over 8},{{3\pi } \...
JEE Advanced 2021 Paper 2 Online
Let $${g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2$$, and $$f:\left[ {{\pi \over 8},{{3\pi } \...
JEE Advanced 2021 Paper 2 Online
Let f1 : (0, $$\infty$$) $$\to$$ R and f2 : (0, $$\infty$$) $$\to$$ R be defined by $${f_1}(x) = \int\limits_0^x {\prod\...
JEE Advanced 2021 Paper 2 Online
Let f1 : (0, $$\infty$$) $$\to$$ R and f2 : (0, $$\infty$$) $$\to$$ R be defined by $${f_1}(x) = \int\limits_0^x {\prod\...
JEE Advanced 2021 Paper 2 Online
Let $$f:R \to R$$ be a differentiable function such that its derivative f' is continuous and f($$\pi $$) = $$-$$6.If $$F...
JEE Advanced 2020 Paper 2 Offline
The value of the integral $$ \int\limits_0^{\pi /2} {{{3\sqrt {\cos \theta } } \over {{{(\sqrt {\cos \theta } + \sqrt {...
JEE Advanced 2019 Paper 2 Offline
If $$I = {2 \over \pi }\int\limits_{ - \pi /4}^{\pi /4} {{{dx} \over {(1 + {e^{\sin x}})(2 - \cos 2x)}}} $$, then 27I2 e...
JEE Advanced 2019 Paper 1 Offline
The value of the integral$$\int_0^{1/2} {{{1\sqrt 3 } \over {{{({{(x + 1)}^2}{{(1 - x)}^6})}^{1/4}}}}dx} $$ is ...........
JEE Advanced 2018 Paper 2 Offline
A farmer F1 has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1) and R(2, 0). From this land, a neigh...
JEE Advanced 2018 Paper 1 Offline
The total number of distinct $$x \in \left[ {0,1} \right]$$ for which $$\int\limits_0^x {{{{t^2}} \over {1 + {t^4}}}} dt...
JEE Advanced 2016 Paper 1 Offline
Let $$F\left( x \right) = \int\limits_x^{{x^2} + {\pi \over 6}} {2{{\cos }^2}t\left( {dt} \right)} $$ for all $$x \in R...
JEE Advanced 2015 Paper 1 Offline
Let $$f:R \to R$$ be a continuous odd function, which vanishes exactly at one point and $$f\left( 1 \right) = {1 \over {...
JEE Advanced 2015 Paper 2 Offline
If $$\alpha = \int\limits_0^1 {\left( {{e^{9x + 3{{\tan }^{ - 1}}x}}} \right)\left( {{{12 + 9{x^2}} \over {1 + {x^2}}}}...
JEE Advanced 2015 Paper 2 Offline
Let $$f:R \to R$$ be a function defined by $$f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2...
JEE Advanced 2015 Paper 1 Offline
The value of $$\int\limits_0^1 {4{x^3}\left\{ {{{{d^2}} \over {d{x^2}}}{{\left( {1 - {x^2}} \right)}^5}} \right\}dx} $$ ...
JEE Advanced 2014 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...
IIT-JEE 2010 Paper 1 Offline
Let $$f:R \to R$$ be a continuous function which satisfies $$$f\left( x \right) = \int\limits_0^x {f\left( t \right)dt....
IIT-JEE 2009

MCQ (Single Correct Answer)

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Which of the following statements is TRUE?
JEE Advanced 2021 Paper 2 Online
Which of the following statements is TRUE?
JEE Advanced 2021 Paper 2 Online
The area of the region $$\left\{ {\matrix{ {(x,y):0 \le x \le {9 \over 4},} & {0 \le y \le 1,} & {x \ge 3y,} ...
JEE Advanced 2021 Paper 1 Online
Let the functions f : R $$ \to $$ R and g : R $$ \to $$ R be defined byf(x) = ex $$-$$ 1 $$-$$ e$$-$$|x $$-$$ 1|and g(x)...
JEE Advanced 2020 Paper 1 Offline
The area of the region{(x, y) : xy $$ \le $$ 8, 1 $$ \le $$ y $$ \le $$ x2} is
JEE Advanced 2019 Paper 1 Offline
Area of the region $$\left\{ {\left( {x,y} \right) \in {R^2}:y \ge \sqrt {\left| {x + 3} \right|} ,5y \le x + 9 \le 15} ...
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 2016 Paper 2 Offline
Let $$f'\left( x \right) = {{192{x^3}} \over {2 + {{\sin }^4}\,\pi x}}$$ for all $$x \in R\,\,$$ with $$f\left( {{1 \ove...
JEE Advanced 2015 Paper 2 Offline
Given that for each $$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{...
JEE Advanced 2014 Paper 2 Offline
Given that for each $$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{...
JEE Advanced 2014 Paper 2 Offline
List - $$I$$ P.$$\,\,\,\,$$ The number of polynomials $$f(x)$$ with non-negative integer coefficients of degree $$ \le 2...
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...
JEE Advanced 2014 Paper 2 Offline
Let $$f$$ $$:\,\,\left[ {{1 \over 2},1} \right] \to R$$ (the set of all real number) be a positive, non-constant and di...
JEE Advanced 2013 Paper 1 Offline
The area enclosed by the curves $$y = \sin x + {\mathop{\rm cosx}\nolimits} $$ and $$y = \left| {\cos x - \sin x} \right...
JEE Advanced 2013 Paper 1 Offline
The value of the integral $$\int\limits_{ - \pi /2}^{\pi /2} {\left( {{x^2} + 1n{{\pi + x} \over {\pi - x}}} \right)\c...
IIT-JEE 2012 Paper 2 Offline
Let f $$:$$$$\left[ { - 1,2} \right] \to \left[ {0,\infty } \right]$$ be a continuous function such that $$f\left( x \r...
IIT-JEE 2011 Paper 2 Offline
Let the straight line $$x=b$$ divide the area enclosed by $$y = {\left( {1 - x} \right)^2},y = 0,$$ and $$x=0$$ into tw...
IIT-JEE 2011 Paper 1 Offline
The value of $$\,\int\limits_{\sqrt {\ell n2} }^{\sqrt {\ell n3} } {{{x\sin {x^2}} \over {\sin {x^2} + \sin \left( {\ell...
IIT-JEE 2011 Paper 1 Offline
Consider the polynomial $$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$ Let $$s$$ be the sum of all distinct real root...
IIT-JEE 2010 Paper 2 Offline
Consider the polynomial $$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$ Let $$s$$ be the sum of all distinct real root...
IIT-JEE 2010 Paper 2 Offline
Consider the polynomial $$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$ Let $$s$$ be the sum of all distinct real root...
IIT-JEE 2010 Paper 2 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 1 Offline
Let $$f$$ be a real-valued function defined on the interval $$(-1, 1)$$ such that $${e^{ - x}}f\left( x \right) = 2 + \...
IIT-JEE 2010 Paper 2 Offline
The value of $$\mathop {\lim }\limits_{x \to 0} {1 \over {{x^3}}}\int\limits_0^x {{{t\ln \left( {1 + t} \right)} \over {...
IIT-JEE 2010 Paper 1 Offline
let $$f$$be a non-negative function defined on the interval $$\left[ {0,1} \right]$$. If $$\int\limits_0^x {\sqrt {1 - ...
IIT-JEE 2009
Consider the function $$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$ defined by $$f\...
IIT-JEE 2008
Consider the function $$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$ defined by $$f\...
IIT-JEE 2008
Consider the function $$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$ defined by $$f\...
IIT-JEE 2008
Consider the functions defined implicitly by the equation $${y^3} - 3y + x = 0$$ on various intervals in the real line...
IIT-JEE 2008
Consider the functions defined implicitly by the equation $${y^3} - 3y + x = 0$$ on various intervals in the real line...
IIT-JEE 2008
Consider the functions defined implicitly by the equation $${y^3} - 3y + x = 0$$ on various intervals in the real line...
IIT-JEE 2008
The area of the region between the curves $$y = \sqrt {{{1 + \sin x} \over {\cos x}}} $$ and $$y = \sqrt {{{1 - \sin x}...
IIT-JEE 2008
Let the definite integral be defined by the formula $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f...
IIT-JEE 2006
Let the definite integral be defined by the formula $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f...
IIT-JEE 2006
Let the definite integral be defined by the formula $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f...
IIT-JEE 2006
$$\int\limits_{ - 2}^0 {\left\{ {{x^3} + 3{x^2} + 3x + 3 + \left( {x + 1} \right)\cos \left( {x + 1} \right)} \right\}\,...
IIT-JEE 2005 Screening
The area bounded by the parabola $$y = {\left( {x + 1} \right)^2}$$ and $$y = {\left( {x - 1} \right)^2}$$ and the line...
IIT-JEE 2005 Screening
If $$f(x)$$ is differentiable and $$\int\limits_0^{{t^2}} {xf\left( x \right)dx = {2 \over 5}{t^5},} $$ then $$f\left( {...
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
The area enclosed between the curves $$y = a{x^2}$$ and $$x = a{y^2}\left( {a > 0} \right)$$ is $$1$$ sq. unit, then...
IIT-JEE 2004 Screening
The area bounded by the curves $$y = \sqrt x ,2y + 3 = x$$ and $$x$$-axis in the 1st quadrant is
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 2003 Screening
If $$l\left( {m,n} \right) = \int\limits_0^1 {{t^m}{{\left( {1 + t} \right)}^n}dt,} $$ then the expression for $$l(m, n)...
IIT-JEE 2003 Screening
Let $$T>0$$ be a fixed real number . Suppose $$f$$ is a continuous function such that for all $$x \in R$$, $$f\left...
IIT-JEE 2002 Screening
Let $$T>0$$ be a fixed real number . Suppose $$f$$ is a continuous function such that for all $$x \in R$$, $$f\left...
IIT-JEE 2002 Screening
Let $$f\left( x \right) = \int\limits_1^x {\sqrt {2 - {t^2}} \,dt.} $$ Then the real roots of the equation $${x^2} - f...
IIT-JEE 2002 Screening
The integral $$\int\limits_{ - 1/2}^{1/2} {\left( {\left[ x \right] + \ell n\left( {{{1 + x} \over {1 - x}}} \right)} \r...
IIT-JEE 2002 Screening
The area bounded by the curves $$y = \left| x \right| - 1$$ and $$y = - \left| x \right| + 1$$ is
IIT-JEE 2002 Screening
The value of $$\int\limits_{ - \pi }^\pi {{{{{\cos }^2}x} \over {1 + {a^x}}}dx,\,a > 0,} $$ is
IIT-JEE 2001 Screening
The value of the integral $$\int\limits_{{e^{ - 1}}}^{{e^2}} {\left| {{{{{\log }_e}x} \over x}} \right|dx} $$ is :
IIT-JEE 2000 Screening
If $$f\left( x \right) = \left\{ {\matrix{ {{e^{\cos x}}\sin x,} & {for\,\,\left| x \right| \le 2} \cr {2,} ...
IIT-JEE 2000 Screening
Let $$g\left( x \right) = \int\limits_0^x {f\left( t \right)dt,} $$ where f is such that $${1 \over 2} \le f\left( t \r...
IIT-JEE 2000 Screening
$$\int\limits_{\pi /4}^{3\pi /4} {{{dx} \over {1 + \cos x}}} $$ is equal to
IIT-JEE 1999
If for a real number $$y$$, $$\left[ y \right]$$ is the greatest integer less than or equal to $$y$$, then the value of...
IIT-JEE 1999
Let $$f\left( x \right) = x - \left[ x \right],$$ for every real number $$x$$, where $$\left[ x \right]$$ is the integra...
IIT-JEE 1998
If $$\int_0^x {f\left( t \right)dt = x + \int_x^1 {t\,\,f\left( t \right)\,\,dt,} } $$ then the value of $$f(1)$$ is
IIT-JEE 1998
If $$g\left( x \right) = \int_0^x {{{\cos }^4}t\,dt,} $$ then $$g\left( {x + \pi } \right)$$ equals
IIT-JEE 1997
If $$f\left( x \right)\,\,\, = \,\,\,A\sin \left( {{{\pi x} \over 2}} \right)\,\,\, + \,\,\,B,\,\,\,f'\left( {{1 \over 2...
IIT-JEE 1995 Screening
The value of $$\int\limits_\pi ^{2\pi } {\left[ {2\,\sin x} \right]\,dx} $$ where [ . ] represents the greatest integer ...
IIT-JEE 1995 Screening
The value of $$\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}} $$ is
IIT-JEE 1993
Let $$f:R \to R$$ and $$\,\,g:R \to R$$ be continuous functions. Then the value of the integral $$\int\limits_{ - \pi...
IIT-JEE 1990
For any integer $$n$$ the integral ........... $$\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right...
IIT-JEE 1985
The value of the integral $$\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx} $$ ...
IIT-JEE 1983
The area bounded by the curves $$y=f(x)$$, the $$x$$-axis and the ordinates $$x=1$$ and $$x=b$$ is $$(b-1)$$ sin $$(3b+4...
IIT-JEE 1982
Let $$a, b, c$$ be non-zero real numbers such that $$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} +...
IIT-JEE 1981
The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$$
IIT-JEE 1981

MCQ (More than One Correct Answer)

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For any real numbers $$\alpha$$ and $$\beta$$, let $${y_{\alpha ,\beta }}(x)$$, x$$\in$$R, be the solution of the differ...
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 $$...
JEE Advanced 2021 Paper 2 Online
Let b be a nonzero real number. Suppose f : R $$ \to $$ R is a differentiable function such that f(0) = 1. If the deriva...
JEE Advanced 2020 Paper 2 Offline
Which of the following inequalities is/are TRUE?
JEE Advanced 2020 Paper 1 Offline
Let f : [0, $$\infty $$) $$ \to $$ R be a continuous function such that$$f(x) = 1 - 2x + \int_0^x {{e^{x - t}}f(t)dt} $$...
JEE Advanced 2018 Paper 1 Offline
If the line x = $$\alpha $$ divides the area of region R = {(x, y) $$ \in $$R2 : x3 $$ \le $$ y $$ \le $$ x, 0 $$ \le $$...
JEE Advanced 2017 Paper 2 Offline
If $$I = \sum\nolimits_{k = 1}^{98} {\int_k^{k + 1} {{{k + 1} \over {x(x + 1)}}} dx} $$, then
JEE Advanced 2017 Paper 2 Offline
Let $$f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \o...
JEE Advanced 2016 Paper 2 Offline
Let $$F:R \to R$$ be a thrice differentiable function. Suppose that $$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$ ...
JEE Advanced 2015 Paper 2 Offline
Let $$F:R \to R$$ be a thrice differentiable function. Suppose that $$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$ ...
JEE Advanced 2015 Paper 2 Offline
The option(s) with the values of a and $$L$$ that satisfy the following equation is (are) $$${{\int\limits_0^{4\pi } {{...
JEE Advanced 2015 Paper 2 Offline
Let $$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...
JEE Advanced 2015 Paper 2 Offline
Let a $$\in$$ R and f : R $$\to$$ R be given by f(x) = x5 $$-$$ 5x + a. Then,
JEE Advanced 2014 Paper 1 Offline
Let $$f:\left( {0,\infty } \right) \to R$$ be given by $$f\left( x \right) = \int\limits_{{1 \over x}}^x {{e^{ - \left( ...
JEE Advanced 2014 Paper 1 Offline
Let $$S$$ be the area of the region enclosed by $$y = {e^{ - {x^2}}}$$, $$y=0$$, $$x=0$$, and $$x=1$$; then
IIT-JEE 2012 Paper 1 Offline
Let $$f$$ be a real-valued function defined on the interval $$\left( {0,\infty } \right)$$ by $$\,f\left( x \right) = \...
IIT-JEE 2010
Let $$f$$ be a real-valued function defined on the interval $$\left( {0,\infty } \right)$$ by $$\,f\left( x \right) = \...
IIT-JEE 2010
Let $$f$$ be a real-valued function defined on the interval $$\left( {0,\infty } \right)$$ by $$\,f\left( x \right) = \...
IIT-JEE 2010
Let $$f$$ be a real-valued function defined on the interval $$\left( {0,\infty } \right)$$ by $$\,f\left( x \right) = \...
IIT-JEE 2010 Paper 1 Offline
If $${I_n} = \int\limits_{ - \pi }^\pi {{{\sin nx} \over {\left( {1 + {\pi ^x}} \right)\sin x}}dx\,\,n = 0,1,2,.....,} ...
IIT-JEE 2009
Area of the region bounded by the curve $$y = {e^x}$$ and lines $$x=0$$ and $$y=e$$ is
IIT-JEE 2009
Let $$f(x)$$ be a non-constant twice differentiable function defined on $$\left( { - \infty ,\infty } \right)$$ such th...
IIT-JEE 2008
For which of the following values of $$m$$, is the area of the region bounded by the curve $$y = x - {x^2}$$ and the lin...
IIT-JEE 1999

Subjective

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Match the integrals in Column $$I$$ with the values in Column $$II$$ and indicate your answer by darkening the appropria...
IIT-JEE 2007
Match the following : Column $$I$$ (A) $$\int\limits_0^{\pi /2} {{{\left( {\sin x} \right)}^{\cos x}}\left( {\cos x\cot ...
IIT-JEE 2006
The value of $$5050{{\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx} \over {\int\limits_0^1 {{{\left( {1 -...
IIT-JEE 2006
If $$\left[ {\matrix{ {4{a^2}} & {4a} & 1 \cr {4{b^2}} & {4b} & 1 \cr {4{c^2}} & {4c} ...
IIT-JEE 2005
$$f(x)$$ is a differentiable function and $$g(x)$$ is a double differentiable function such that $$\left| {f\left( x \r...
IIT-JEE 2005
Find the area bounded by the curves $${x^2} = y,{x^2} = - y$$ and $${y^2} = 4x - 3.$$
IIT-JEE 2005
Evaluate $$\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos...
IIT-JEE 2005
Find the value of $$\int\limits_{ - \pi /3}^{\pi /3} {{{\pi + 4{x^3}} \over {2 - \cos \left( {\left| x \right| + {\pi ...
IIT-JEE 2004
If $$y\left( x \right) = \int\limits_{{x^2}/16}^{{x^2}} {{{\cos x\cos \sqrt \theta } \over {1 + {{\sin }^2}\sqrt \theta...
IIT-JEE 2004
If $$f$$ is an even function then prove that $$\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } ...
IIT-JEE 2003
Find the area of the region bounded by the curves $$y = {x^2},y = \left| {2 - {x^2}} \right|$$ and $$y=2,$$ which lies t...
IIT-JEE 2002
Let $$b \ne 0$$ and for $$j=0, 1, 2, ..., n,$$ let $${S_j}$$ be the area of the region bounded by the $$y$$-axis and t...
IIT-JEE 2001
For $$x>0,$$ let $$f\left( x \right) = \int\limits_e^x {{{\ln t} \over {1 + t}}dt.} $$ Find the function $$f\left( x...
IIT-JEE 2000
Let $$f(x)$$ be a continuous function given by $$$f\left( x \right) = \left\{ {\matrix{ {2x,} & {\left| x \right...
IIT-JEE 1999
Integrate $$\int\limits_0^\pi {{{{e^{\cos x}}} \over {{e^{\cos x}} + {e^{ - \cos x}}}}\,dx.} $$
IIT-JEE 1999
Prove that $$\int_0^1 {{{\tan }^{ - 1}}} \,\left( {{1 \over {1 - x + {x^2}}}} \right)dx = 2\int_0^1 {{{\tan }^{ - 1}}} \...
IIT-JEE 1998
Determine the value of $$\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.$$
IIT-JEE 1997
Let $$f(x)= Maximum $$ $$\,\left\{ {{x^2},{{\left( {1 - x} \right)}^2},2x\left( {1 - x} \right)} \right\},$$ where $$0 ...
IIT-JEE 1997
Let $${A_n}$$ be the area bounded by the curve $$y = {\left( {\tan x} \right)^n}$$ and the lines $$x=0,$$ $$y=0,$$ and ...
IIT-JEE 1996
Evaluate the definite integral : $$$\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \righ...
IIT-JEE 1995
Consider a square with vertices at $$(1,1), (-1,1), (-1,-1)$$ and $$(1, -1)$$. Let $$S$$ be the region consisting of all...
IIT-JEE 1995
Let $${I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.$$ Use mathematical induction to prove that $${...
IIT-JEE 1995
In what ratio does the $$x$$-axis divide the area of the region bounded by the parabolas $$y = 4x - {x^2}$$ and $$y = {...
IIT-JEE 1994
Show that $$\int\limits_0^{n\pi + v} {\left| {\sin x} \right|dx = 2n + 1 - \cos \,v} $$ where $$n$$ is a positive integ...
IIT-JEE 1994
Evaluate $$\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)...
IIT-JEE 1993
Determine a positive integer $$n \le 5,$$ such that $$$\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e} ...
IIT-JEE 1992
Sketch the region bounded by the curves $$y = {x^2}$$ and $$y = {2 \over {1 + {x^2}}}.$$ Find the area.
IIT-JEE 1992
If $$'f$$ is a continuous function with $$\int\limits_0^x {f\left( t \right)dt \to \infty } $$ as $$\left| x \right| \to...
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 1991
Sketch the curves and identify the region bounded by $$x = {1 \over 2},x = 2,y = \ln \,x$$ and $$y = {2^x}.$$ Find the ...
IIT-JEE 1991
Show that $$\int\limits_0^{\pi /2} {f\left( {\sin 2x} \right)\sin x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\co...
IIT-JEE 1990
Compute the area of the region bounded by the curves $$\,y = ex\,\ln x$$ and $$y = {{\ln x} \over {ex}}$$ where $$ln$$ $...
IIT-JEE 1990
Prove that for any positive integer $$k$$, $${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos ...
IIT-JEE 1990
If $$f$$ and $$g$$ are continuous function on $$\left[ {0,a} \right]$$ satisfying $$f\left( x \right) = f\left( {a - x}...
IIT-JEE 1989
Evaluate $$\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx} $$
IIT-JEE 1988
Find the area of the region bounded by the curve $$C:y=$$ $$\tan x,$$ tangent drawn to $$C$$ at $$x = {\pi \over 4}$$ ...
IIT-JEE 1988
Find the area bounded by the curves, $${x^2} + {y^2} = 25,\,4y = \left| {4 - {x^2}} \right|$$ and $$x=0$$ above the $$x$...
IIT-JEE 1987
Evaluate : $$\int\limits_0^\pi {{{x\,dx} \over {1 + \cos \,\alpha \,\sin x}},0 < \alpha < \pi } $$
IIT-JEE 1986
Evaluate the following : $$\,\,\int\limits_0^{\pi /2} {{{x\sin x\cos x} \over {{{\cos }^4}x + {{\sin }^4}x}}} dx$$
IIT-JEE 1985
Sketch the region bounded by the curves $$y = \sqrt {5 - {x^2}} $$ and $$y = \left| {x - 1} \right|$$ and find its area.
IIT-JEE 1985
Find the area of the region bounded by the $$x$$-axis and the curves defined by $$$y = \tan x, - {\pi \over 3} \le x \...
IIT-JEE 1984
Given a function $$f(x)$$ such that (i) it is integrable over every interval on the real line and (ii) $$f(t+x)=f(x),$...
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
Find the area bounded by the $$x$$-axis, part of the curve $$y = \left( {1 + {8 \over {{x^2}}}} \right)$$ and the ordin...
IIT-JEE 1983
Evaluate : $$\int\limits_0^{\pi /4} {{{\sin x + \cos x} \over {9 + 16\sin 2x}}dx} $$
IIT-JEE 1983
For any real $$t,\,x = {{{e^t} + {e^{ - t}}} \over 2},\,\,y = {{{e^t} - {e^{ - t}}} \over 2}$$ is a point on the hyperb...
IIT-JEE 1982
Show that $$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} = {\pi \over 2}\int\limits_0^\pi {f\left( {\sin x} \righ...
IIT-JEE 1982
Find the value of $$\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx} $$
IIT-JEE 1982
Show that : $$\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n...
IIT-JEE 1981
Find the area bounded by the curve $${x^2} = 4y$$ and the straight
IIT-JEE 1981

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Let $${d \over {dx}}\,F\left( x \right) = {{{e^{\sin x}}} \over x},\,x > 0.$$ If $$\int_1^4 {{{2{e^{\sin {x^2}}}} \ov...
IIT-JEE 1997
The value of $$\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx} $$ is ...............
IIT-JEE 1997
For $$n>0,$$ $$\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx = $$
IIT-JEE 1996
If for nonzero $$x$$, $$af(x)+$$ $$bf\left( {{1 \over x}} \right) = {1 \over x} - 5$$ where $$a \ne b,$$ then $$\int_1^...
IIT-JEE 1996
The value of $$\int\limits_2^3 {{{\sqrt x } \over {\sqrt {3 - x} + \sqrt x }}} dx$$ is ...........
IIT-JEE 1994
The value of $$\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi } $$ is ..............
IIT-JEE 1993
The value of $$\int\limits_{ - 2}^2 {\left| {1 - {x^2}} \right|dx} $$ is ...............
IIT-JEE 1989
The integral $$\int\limits_0^{1.5} {\left[ {{x^2}} \right]dx,} $$ Where [ ] denotes the greatest integer function, equa...
IIT-JEE 1988
$$f\left( x \right) = \left| {\matrix{ {\sec x} & {\cos x} & {{{\sec }^2}x + \cot x\cos ec\,x} \cr {{{\c...
IIT-JEE 1987

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The value of the integral $$\int\limits_0^{2a} {[{{f\left( x \right)} \over {\left\{ {f\left( x \right) + f\left( {2a - ...
IIT-JEE 1988

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