# Definite Integrals and Applications of Integrals · Mathematics · JEE Advanced

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## MCQ (Single Correct Answer)

JEE Advanced 2024 Paper 2 Online
Let $S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0, y \geq 0, y^2 \leq 4 x, y^2 \leq 12-2 x\right.$ and $\left.3 y+\sqrt{8} x \leq 5 \sqr... JEE Advanced 2023 Paper 1 Online Let$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 2021 Paper 1 Online
The area of the region $$\left\{ {\matrix{ {(x,y):0 \le x \le {9 \over 4},} & {0 \le y \le 1,} & {x \ge 3y,} & {x + y \ge 2} \cr } ... JEE Advanced 2020 Paper 1 Offline Let the functions f : R$$ \to $$R and g : R$$ \to $$R be defined byf(x) = ex$$-$$1$$-$$e$$-$$|x$$-$$1|and g(x) =$${1 \over 2}$$(ex$$-$$1 ... JEE Advanced 2019 Paper 1 Offline The area of the region{(x, y) : xy$$ \le $$8, 1$$ \le $$y$$ \le $$x2} is JEE Advanced 2016 Paper 2 Offline The value of$$\int\limits_{-{\pi \over 2}}^{{\pi \over 2}} {{{{x^2}\cos x} \over {1 + {e^x}}}dx} $$is equal to JEE Advanced 2016 Paper 2 Offline Area of the region$$\left\{ {\left( {x,y} \right) \in {R^2}:y \ge \sqrt {\left| {x + 3} \right|} ,5y \le x + 9 \le 15} \right\}$$is equal to ... JEE Advanced 2015 Paper 2 Offline Let$$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
The area enclosed by the curves $$y = \sin x + {\mathop{\rm cosx}\nolimits}$$ and $$y = \left| {\cos x - \sin x} \right|$$ over the interval $$\left[... JEE Advanced 2013 Paper 1 Offline Let$$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} $$... IIT-JEE 2011 Paper 1 Offline Let the straight line$$x=b$$divide the area enclosed by$$y = {\left( {1 - x} \right)^2},y = 0,$$and$$x=0$$into two parts$${R_1}\left( {0 \le x...
IIT-JEE 2011 Paper 2 Offline
Let f $$:$$$$\left[ { - 1,2} \right] \to \left[ {0,\infty } \right]$$ be a continuous function such that $$f\left( x \right) = f\left( {1 - x} \right... IIT-JEE 2010 Paper 1 Offline The value of$$\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 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 roots of $$f(x)$$ and let $$t = \l... IIT-JEE 2010 Paper 2 Offline Consider the polynomial$$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$Let$$s$$be the sum of all distinct real roots of$$f(x)$$and let$$t = \l...
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 roots of $$f(x)$$ and let $$t = \l... IIT-JEE 2009 Paper 1 Offline Let$$f$$be a non-negative function defined on the interval$$[0,1]$$. If$$\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}dt} = \int\limits_0^x {f(t)dt,...
IIT-JEE 2008 Paper 2 Offline
The area of the region between the curves $$y = \sqrt {{{1 + \sin x} \over {\cos x}}}$$ and $$y = \sqrt {{{1 - \sin x} \over {\cos x}}}$$ bounded b...
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
The area of the region bounded by the curve $$y=f(x),$$ the $$x$$-axis, and the lines $$x=a$$ and $$x=b$$, where $$- \infty < a < b < - 2,... IIT-JEE 2008 Paper 1 Offline$$\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
The area bounded by the parabola $$y = {\left( {x + 1} \right)^2}$$ and $$y = {\left( {x - 1} \right)^2}$$ and the line $$y=1/4$$ is
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 area enclosed between the curves $$y = a{x^2}$$ and $$x = a{y^2}\left( {a > 0} \right)$$ is $$1$$ sq. unit, then the value of $$a$$ is
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 2003 Screening
The area bounded by the curves $$y = \sqrt x ,2y + 3 = x$$ and $$x$$-axis in the 1st quadrant is
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 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 $$f\left( x \right) = \int\limits_1^x {\sqrt {2 - {t^2}} \,dt.}$$ Then the real roots of the equation $${x^2} - f'\left( x \right) = 0$$ are ...
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
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 $$t \in \le... IIT-JEE 2000 Screening If$$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
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 1997
If $$g\left( x \right) = \int_0^x {{{\cos }^4}t\,dt,}$$ then $$g\left( {x + \pi } \right)$$ equals
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} $$is IIT-JEE 1982 The area bounded by the curves$$y=f(x)$$, the$$x$$-axis and the ordinates$$x=1$$and$$x=b$$is$$(b-1)$$sin$$(3b+4)$$. Then$$f(x)$$is IIT-JEE 1981 The value of the definite integral$$\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... ## Numerical JEE Advanced 2024 Paper 2 Online Let the function$f:[1, \infty) \rightarrow \mathbb{R}$be defined by $$f(t)=\left\{\begin{array}{cc} (-1)^{n+1} 2, & \text { if } t=2 n-1, n \in \m... JEE Advanced 2024 Paper 2 Online The value of 2 \int\limits_0^{\frac{\pi}{2}} f(x) g(x) d x-\int\limits_0^{\frac{\pi}{2}} g(x) d x is ____________. JEE Advanced 2024 Paper 2 Online The value of \frac{16}{\pi^3} \int\limits_0^{\frac{\pi}{2}} f(x) g(x) d x is ______. JEE Advanced 2023 Paper 2 Online For x \in \mathbb{R}, let \tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right). Then the minimum value of the function f: \mathbb{R} \righ... JEE Advanced 2023 Paper 1 Online Let n \geq 2 be a natural number and f:[0,1] \rightarrow \mathbb{R} be the function defined by$$ f(x)= \begin{cases}n(1-2 n x) & \text { if } 0 \... JEE Advanced 2022 Paper 2 Online The greatest integer less than or equal to $$\int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}}... JEE Advanced 2022 Paper 2 Online Consider the functions f, g: \mathbb{R} \rightarrow \mathbb{R} defined by$$ f(x)=x^{2}+\frac{5}{12} \quad \text { and } \quad g(x)= \begin{cases}2... JEE Advanced 2021 Paper 2 Online Let f1 : (0, $$\infty$$) $$\to$$ R and f2 : (0, $$\infty$$) $$\to$$ R be defined by $${f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)... JEE Advanced 2021 Paper 2 Online Let f1 : (0,$$\infty$$)$$\to$$R and f2 : (0,$$\infty$$)$$\to$$R be defined by$${f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)... 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 2018 Paper 1 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 neighbouring farmer F2 takes away t... 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 $${\tan ^... JEE Advanced 2015 Paper 2 Offline Let$$f:R \to R$$be a continuous odd function, which vanishes exactly at one point and$$f\left( 1 \right) = {1 \over {2.}}$$Suppose that$$F\left( ... JEE Advanced 2015 Paper 1 Offline Let $$f:R \to R$$ be a function defined by $$f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2} \cr {0,} & {x > ... JEE Advanced 2015 Paper 1 Offline Let$$F\left( x \right) = \int\limits_x^{{x^2} + {\pi \over 6}} {2{{\cos }^2}t\left( {dt} \right)} $$for all$$x \in R$$and$$f:\left[ {0,{1 \over ... 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 (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 2021 Paper 2 Online For any real numbers $$\alpha$$ and $$\beta$$, let $${y_{\alpha ,\beta }}(x)$$, x$$\in$$R, be the solution of the differential equation $${{dy} \over ... JEE Advanced 2020 Paper 2 Offline Let b be a nonzero real number. Suppose f : R$$ \to $$R is a differentiable function such that f(0) = 1. If the derivative f' of f satisfies the equ... JEE Advanced 2020 Paper 1 Offline Which of the following inequalities is/are TRUE? JEE Advanced 2018 Paper 1 Offline Let f : [0,$$\infty $$)$$ \to $$R be a continuous function such that$$f(x) = 1 - 2x + \int_0^x {{e^{x - t}}f(t)dt} $$for all x$$ \in $$[0,$$\in... 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 If the line x = $$\alpha$$ divides the area of region R = {(x, y) $$\in$$R2 : x3 $$\le$$ y $$\le$$ x, 0 $$\le$$ x $$\le$$ 1} into two equal... 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 2015 Paper 2 Offline
Let $$F:R \to R$$ be a thrice differentiable function. Suppose that $$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$ and $$F\left( x \right) < 0... JEE Advanced 2015 Paper 2 Offline Let$$F:R \to R$$be a thrice differentiable function. Suppose that$$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$and$$F'\left( x \right) < ...
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$$R be given by f(x) = x5$$-$$5x + a. Then, IIT-JEE 2012 Paper 1 Offline Let$$S$$be the area of the region enclosed by$$y = {e^{ - {x^2}}}$$,$$y=0$$,$$x=0$$, and$$x=1$$; then IIT-JEE 2010 Paper 1 Offline Let$$f$$be a real-valued function defined on the interval$$\left( {0,\infty } \right)$$by$$\,f\left( x \right) = \ln x + \int\limits_0^x {\sqrt ...
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
IIT-JEE 2009 Paper 1 Offline
Area of the region bounded by the curve $$y = {e^x}$$ and lines $$x=0$$ and $$y=e$$ is
IIT-JEE 1999
For which of the following values of $$m$$, is the area of the region bounded by the curve $$y = x - {x^2}$$ and the line $$y=mx$$ equals $$9/2$$?

## 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 2006
Match the following : Column $$I$$ (A) $$\int\limits_0^{\pi /2} {{{\left( {\sin x} \right)}^{\cos x}}\left( {\cos x\cot x - \log {{\left( {\sin x} \ri... IIT-JEE 2005 Evaluate$$\,\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 2005
Find the area bounded by the curves $${x^2} = y,{x^2} = - y$$ and $${y^2} = 4x - 3.$$
IIT-JEE 2005
$$f(x)$$ is a differentiable function and $$g(x)$$ is a double differentiable function such that $$\left| {f\left( x \right)} \right| \le 1$$ and $$f... IIT-JEE 2005 If$$\left[ {\matrix{ {4{a^2}} & {4a} & 1 \cr {4{b^2}} & {4b} & 1 \cr {4{c^2}} & {4c} & 1 \cr } } \right]\le...
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 2002
Find the area of the region bounded by the curves $$y = {x^2},y = \left| {2 - {x^2}} \right|$$ and $$y=2,$$ which lies to the right of the line $$x=1.... IIT-JEE 2001 Let$$b \ne 0$$and for$$j=0, 1, 2, ..., n,$$let$${S_j}$$be the area of the region bounded by the$$y$$-axis and the curve$$x{e^{ay}} = \sin $$... IIT-JEE 2000 For$$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 1999
Let $$f(x)$$ be a continuous function given by $$f\left( x \right) = \left\{ {\matrix{ {2x,} & {\left| x \right| \le 1} \cr {{x^2} + ax ... IIT-JEE 1998 Prove that$$\int_0^1 {{{\tan }^{ - 1}}} \,\left( {{1 \over {1 - x + {x^2}}}} \right)dx = 2\int_0^1 {{{\tan }^{ - 1}}} \,x\,dx.$$Hence or otherwise, ... IIT-JEE 1997 Let$$f(x)= Maximum \,\left\{ {{x^2},{{\left( {1 - x} \right)}^2},2x\left( {1 - x} \right)} \right\},$$where$$0 \le x \le 1.$$Determine the ... IIT-JEE 1997 Determine the value of$$\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.$$IIT-JEE 1996 Let$${A_n}$$be the area bounded by the curve$$y = {\left( {\tan x} \right)^n}$$and the lines$$x=0,y=0,$$and$$x = {\pi \over 4}.$$Prove ... IIT-JEE 1995 Let$${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
Consider a square with vertices at $$(1,1), (-1,1), (-1,-1)$$ and $$(1, -1)$$. Let $$S$$ be the region consisting of all points inside the square whic...
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 1994 In what ratio does the $$x$$-axis divide the area of the region bounded by the parabolas $$y = 4x - {x^2}$$ and $$y = {x^2} - x?$$ 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 Sketch the region bounded by the curves $$y = {x^2}$$ and $$y = {2 \over {1 + {x^2}}}.$$ Find the area. 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
Sketch the curves and identify the region bounded by $$x = {1 \over 2},x = 2,y = \ln \,x$$ and $$y = {2^x}.$$ Find the area of this region.
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
If $$'f$$ is a continuous function with $$\int\limits_0^x {f\left( t \right)dt \to \infty }$$ as $$\left| x \right| \to \infty ,$$ then show that eve...
IIT-JEE 1990
Prove that for any positive integer $$k$$, $${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \rig... IIT-JEE 1990 Compute the area of the region bounded by the curves$$\,y = ex\,\ln x$$and$$y = {{\ln x} \over {ex}}$$where$$lne=1.$$IIT-JEE 1990 Show that$$\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
Find the area of the region bounded by the curve $$C:y=$$ $$\tan x,$$ tangent drawn to $$C$$ at $$x = {\pi \over 4}$$ and the $$x$$-axis.
IIT-JEE 1988
Evaluate $$\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx}$$
IIT-JEE 1987
Find the area bounded by the curves, $${x^2} + {y^2} = 25,\,4y = \left| {4 - {x^2}} \right|$$ and $$x=0$$ above the $$x$$-axis.
IIT-JEE 1986
Evaluate : $$\int\limits_0^\pi {{{x\,dx} \over {1 + \cos \,\alpha \,\sin x}},0 < \alpha < \pi }$$
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
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 1984 Find the area of the region bounded by the $$x$$-axis and the curves defined by $$y = \tan x, - {\pi \over 3} \le x \le {\pi \over 3};\,\,y = \cot... IIT-JEE 1983 Evaluate :$$\int\limits_0^{\pi /4} {{{\sin x + \cos x} \over {9 + 16\sin 2x}}dx} $$IIT-JEE 1983 Find the area bounded by the$$x$$-axis, part of the curve$$y = \left( {1 + {8 \over {{x^2}}}} \right)$$and the ordinates at$$x=2$$and$$x=4$$. I... IIT-JEE 1982 Find the value of$$\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,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.} $$IIT-JEE 1982 For any real$$t,\,x = {{{e^t} + {e^{ - t}}} \over 2},\,\,y = {{{e^t} - {e^{ - t}}} \over 2}$$is a point on the hyperbola$${x^2} - {y^2} = 1$$. Sho... IIT-JEE 1981 Find the area bounded by the curve$${x^2} = 4y$$and the straight IIT-JEE 1981 Show that :$$\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,} $$Where [ ] denotes the greatest integer function, equals ............. IIT-JEE 1987$$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}... ## True or False 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}$...
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