Mathematics
Definite Integrals and Applications of Integrals
Previous Years Questions

## Numerical

The greatest integer less than or equal to $$\int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}}... Consider the functions f, g: \mathbb{R} \rightarrow \mathbb{R} defined by$$ f(x)=x^{2}+\frac{5}{12} \quad \text { and } \quad g(x)= \begin{cases}2...
Let f1 : (0, $$\infty$$) $$\to$$ R and f2 : (0, $$\infty$$) $$\to$$ R be defined by $${f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)... Let f1 : (0,$$\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)...
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...
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...
For any real number x, let [ x ] denote the largest integer less than or equal to x. If $$I = \int\limits_0^{10} {\left[ {\sqrt {{{10x} \over {x + 1}}... Let$$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$$is defined b... The value of the integral$$ \int\limits_0^{\pi /2} {{{3\sqrt {\cos \theta } } \over {{{(\sqrt {\cos \theta } + \sqrt {\sin \theta } )}^5}}}} d\theta...
If $$I = {2 \over \pi }\int\limits_{ - \pi /4}^{\pi /4} {{{dx} \over {(1 + {e^{\sin x}})(2 - \cos 2x)}}}$$, then 27I2 equals ....................
The value of the integral$$\int_0^{1/2} {{{1\sqrt 3 } \over {{{({{(x + 1)}^2}{{(1 - x)}^6})}^{1/4}}}}dx}$$ is ........
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...
The total number of distinct $$x \in \left[ {0,1} \right]$$ for which $$\int\limits_0^x {{{{t^2}} \over {1 + {t^4}}}} dt = 2x - 1$$
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 $${\tan ^... Let$$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( ...
Let $$f:R \to R$$ be a function defined by $$f\left( x \right) = \left\{ {\matrix{ {\left[ x \right],} & {x \le 2} \cr {0,} & {x > ... Let$$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 ...
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
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...
Let $$f:R \to R$$ be a continuous function which satisfies $$f\left( x \right) = \int\limits_0^x {f\left( t \right)dt.}$$$Then the value of $$f\le... ## MCQ (More than One Correct Answer) Consider the equation$$ \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}} ... Let $$f:\left[ { - {\pi \over 2},{\pi \over 2}} \right] \to R$$ be a continuous function such that $$f(0) = 1$$ and $$\int_0^{{\pi \over 3}} {f(t)d... For any real numbers$$\alpha$$and$$\beta$$, let$${y_{\alpha ,\beta }}(x)$$, x$$\in$$R, be the solution of the differential equation$${{dy} \over ... 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... Which of the following inequalities is/are TRUE? 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, $$\in... If$$I = \sum\nolimits_{k = 1}^{98} {\int_k^{k + 1} {{{k + 1} \over {x(x + 1)}}} dx} $$, then If the line x =$$\alpha $$divides the area of region R = {(x, y)$$ \in $$R2 : x3$$ \le $$y$$ \le $$x, 0$$ \le $$x$$ \le $$1} into two equal... Let$$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 + ... Let $$f\left( x \right) = 7{\tan ^8}x + 7{\tan ^6}x - 3{\tan ^4}x - 3{\tan ^2}x$$ for all $$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right).$$... The option(s) with the values of a and $$L$$ that satisfy the following equation is (are) $${{\int\limits_0^{4\pi } {{e^t}\left( {{{\sin }^6}at + {{... Let$$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... Let $$F:R \to R$$ be a thrice differentiable function. Suppose that $$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$ and $$F\left( x \right) < 0... Let$$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){{dt}... Let a $$\in$$ R and f : R $$\to$$ R be given by f(x) = x5 $$-$$ 5x + a. Then, Let $$S$$ be the area of the region enclosed by $$y = {e^{ - {x^2}}}$$, $$y=0$$, $$x=0$$, and $$x=1$$; then Let $$f$$ be a real-valued function defined on the interval $$\left( {0,\infty } \right)$$ by $$\,f\left( x \right) = \ln x + \int\limits_0^x {\sqrt ... Area of the region bounded by the curve$$y = {e^x}$$and lines$$x=0$$and$$y=e$$is If$${I_n} = \int\limits_{ - \pi }^\pi {{{\sin nx} \over {\left( {1 + {\pi ^x}} \right)\sin x}}dx\,\,n = 0,1,2,.....,} $$then Let$$f(x)$$be a non-constant twice differentiable function defined on$$\left( { - \infty ,\infty } \right)$$such that$$f\left( x \right) = f\lef... 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 $$9/2$$? ## MCQ (Single Correct Answer) Which of the following statements is TRUE? Which of the following statements is TRUE? The area of the region $$\left\{ {\matrix{ {(x,y):0 \le x \le {9 \over 4},} & {0 \le y \le 1,} & {x \ge 3y,} & {x + y \ge 2} \cr } ... Let the functions f : R$$ \to $$R and g : R$$ \to $$R be defined byf(x) = ex$$-$$1$$-$$e$$-$$|x$$-$$1|and g(x) =$${1 \over 2}$$(ex$$-$$1 ... The area of the region{(x, y) : xy$$ \le $$8, 1$$ \le $$y$$ \le $$x2} is The value of$$\int\limits_{-{\pi \over 2}}^{{\pi \over 2}} {{{{x^2}\cos x} \over {1 + {e^x}}}dx} $$is equal to Area of the region$$\left\{ {\left( {x,y} \right) \in {R^2}:y \ge \sqrt {\left| {x + 3} \right|} ,5y \le x + 9 \le 15} \right\}$$is equal to ... Let$$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... The following integral $$\int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\left( {2\cos ec\,\,x} \right)}^{17}}dx}$$ is equal to List - $$I$$ P.$$\,\,\,\,$$ The number of polynomials $$f(x)$$ with non-negative integer coefficients of degree $$\le 2$$, satisfying $$f(0)=0$$ and ... Given that for each $$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \rig... Given that for each$$a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \rig... The area enclosed by the curves $$y = \sin x + {\mathop{\rm cosx}\nolimits}$$ and $$y = \left| {\cos x - \sin x} \right|$$ over the interval $$\left[... Let$$f:\,\,\left[ {{1 \over 2},1} \right] \to R$$(the set of all real number) be a positive, non-constant and differentiable function such tha... The value of the integral$$\int\limits_{ - \pi /2}^{\pi /2} {\left( {{x^2} + 1n{{\pi + x} \over {\pi - x}}} \right)\cos xdx} $$is The value of$$\,\int\limits_{\sqrt {\ell n2} }^{\sqrt {\ell n3} } {{{x\sin {x^2}} \over {\sin {x^2} + \sin \left( {\ell n6 - {x^2}} \right)}}\,dx} $$... Let the straight line$$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... Let f $$:$$$$\left[ { - 1,2} \right] \to \left[ {0,\infty } \right]$$ be a continuous function such that $$f\left( x \right) = f\left( {1 - x} \right... The value of$$\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$$is The value of$$\int\limits_0^1 {{{{x^4}{{\left( {1 - x} \right)}^4}} \over {1 + {x^2}}}dx} $$is (are) Let$$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} +... 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 $$t = \l... Consider the polynomial$$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... 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 $$t = \l... let$$f$$be a non-negative function defined on the interval$$\left[ {0,1} \right]$$. If$$\int\limits_0^x {\sqrt {1 - {{\left( {f'\left( t \right)} ... 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... Consider the functions defined implicitly by the equation $${y^3} - 3y + x = 0$$ on various intervals in the real line. If $$x \in \left( { - \inft... Consider the functions defined implicitly by the equation$${y^3} - 3y + x = 0$$on various intervals in the real line. If$$x \in \left( { - \inft... Consider the functions defined implicitly by the equation $${y^3} - 3y + x = 0$$ on various intervals in the real line. If $$x \in \left( { - \inft... Consider the function$$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$defined by$$f\left( x \right) = {{{x^2} - ax... Consider the function $$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$ defined by $$f\left( x \right) = {{{x^2} - ax... Consider the function$$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$defined by$$f\left( x \right) = {{{x^2} - ax... Let the definite integral be defined by the formula $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \... Let the definite integral be defined by the formula$$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \... Let the definite integral be defined by the formula $$\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \... The area bounded by the parabola$$y = {\left( {x + 1} \right)^2}$$and$$y = {\left( {x - 1} \right)^2}$$and the line$$y=1/4$$is$$\int\limits_{ - 2}^0 {\left\{ {{x^3} + 3{x^2} + 3x + 3 + \left( {x + 1} \right)\cos \left( {x + 1} \right)} \right\}\,\,dx} $$is equal to The area enclosed between the curves$$y = a{x^2}$$and$$x = a{y^2}\left( {a > 0} \right)$$is$$1$$sq. unit, then the value of$$a$$is The value of the integral$$\int\limits_0^1 {\sqrt {{{1 - x} \over {1 + x}}} dx} $$is If$$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)$$equ... If$$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)$$... If$$f\left( x \right) = \int\limits_{{x^2}}^{{x^2} + 1} {{e^{ - {t^2}}}} dt,$$then$$f(x)$$increases in The area bounded by the curves$$y = \sqrt x ,2y + 3 = x$$and$$x$$-axis in the 1st quadrant is The area bounded by the curves$$y = \left| x \right| - 1$$and$$y = - \left| x \right| + 1$$is The integral$$\int\limits_{ - 1/2}^{1/2} {\left( {\left[ x \right] + \ell n\left( {{{1 + x} \over {1 - x}}} \right)} \right)dx} $$equal to Let$$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$$are ... Let$$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 ... Let $$T>0$$ be a fixed real number . Suppose $$f$$ is a continuous function such that for all $$x \in R$$, $$f\left( {x + T} \right) = f\left( x ... The value of$$\int\limits_{ - \pi }^\pi {{{{{\cos }^2}x} \over {1 + {a^x}}}dx,\,a > 0,} $$is Let$$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... If $$f\left( x \right) = \left\{ {\matrix{ {{e^{\cos x}}\sin x,} & {for\,\,\left| x \right| \le 2} \cr {2,} & {otherwise,} \cr } }... The value of the integral$$\int\limits_{{e^{ - 1}}}^{{e^2}} {\left| {{{{{\log }_e}x} \over x}} \right|dx} $$is : If for a real number$$y$$,$$\left[ y \right]$$is the greatest integer less than or equal to$$y$$, then the value of the integral$$\int\limits_{\... $$\int\limits_{\pi /4}^{3\pi /4} {{{dx} \over {1 + \cos x}}}$$ is equal to 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 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 $$\int_{... If$$g\left( x \right) = \int_0^x {{{\cos }^4}t\,dt,} $$then$$g\left( {x + \pi } \right)$$equals The value of$$\int\limits_\pi ^{2\pi } {\left[ {2\,\sin x} \right]\,dx} $$where [ . ] represents the greatest integer function is If$$f\left( x \right)\,\,\, = \,\,\,A\sin \left( {{{\pi x} \over 2}} \right)\,\,\, + \,\,\,B,\,\,\,f'\left( {{1 \over 2}} \right) = \sqrt 2 $$and ... The value of$$\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}} $$is Let$$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(... For any integer $$n$$ the integral ........... $$\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right)xdx}$$ has the value The value of the integral $$\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx}$$ is 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 The value of the definite integral $$\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx$$ Let $$a, b, c$$ be non-zero real numbers such that $$\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limi... ## Subjective Match the integrals in Column$$I$$with the values in Column$$II$$and indicate your answer by darkening the appropriate bubbles in the$$4 \times 4... The value of $$5050{{\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx} \over {\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx... Match the following : Column$$I$$(A)$$\int\limits_0^{\pi /2} {{{\left( {\sin x} \right)}^{\cos x}}\left( {\cos x\cot x - \log {{\left( {\sin x} \ri... Evaluate $$\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos \left( {{1 \over 2}\cos x} \r... Find the area bounded by the curves$${x^2} = y,{x^2} = - y$$and$${y^2} = 4x - 3.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... If $$\left[ {\matrix{ {4{a^2}} & {4a} & 1 \cr {4{b^2}} & {4b} & 1 \cr {4{c^2}} & {4c} & 1 \cr } } \right]\le... If$$y\left( x \right) = \int\limits_{{x^2}/16}^{{x^2}} {{{\cos x\cos \sqrt \theta } \over {1 + {{\sin }^2}\sqrt \theta }}d\theta ,} $$then find$$... Find the value of $$\int\limits_{ - \pi /3}^{\pi /3} {{{\pi + 4{x^3}} \over {2 - \cos \left( {\left| x \right| + {\pi \over 3}} \right)}}dx}$$ If $$f$$ is an even function then prove that $$\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left... Find the area of the region bounded by the curves$$y = {x^2},y = \left| {2 - {x^2}} \right|$$and$$y=2,$$which lies to the right of the line$$x=1.... 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 $$x{e^{ay}} = \sin$$... For $$x>0,$$ let $$f\left( x \right) = \int\limits_e^x {{{\ln t} \over {1 + t}}dt.}$$ Find the function $$f\left( x \right) + f\left( {{1 \over x... Integrate$$\int\limits_0^\pi {{{{e^{\cos x}}} \over {{e^{\cos x}} + {e^{ - \cos x}}}}\,dx.} $$Let$$f(x)$$be a continuous function given by$$$f\left( x \right) = \left\{ {\matrix{ {2x,} & {\left| x \right| \le 1} \cr {{x^2} + ax ...
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, ...
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 ...
Determine the value of $$\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.$$
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 ...
Let $${I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.$$ Use mathematical induction to prove that $${I_m} = m\,\pi ,m = 0,1,2,........ Consider a square with vertices at$$(1,1), (-1,1), (-1,-1)$$and$$(1, -1)$$. Let$$S$$be the region consisting of all points inside the square whic... Evaluate the definite integral :$$$\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \right){{\cos }^{ - 1}}\left( {{{2x... Show that $$\int\limits_0^{n\pi + v} {\left| {\sin x} \right|dx = 2n + 1 - \cos \,v}$$ where $$n$$ is a positive integer and $$\,0 \le v < \pi .... In what ratio does the$$x$$-axis divide the area of the region bounded by the parabolas$$y = 4x - {x^2}$$and$$y = {x^2} - x?$$Evaluate$$\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.$$Sketch the region bounded by the curves$$y = {x^2}$$and$$y = {2 \over {1 + {x^2}}}.$$Find the area. Determine a positive integer$$n \le 5,$$such that$$$\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e} $$Sketch the curves and identify the region bounded by$$x = {1 \over 2},x = 2,y = \ln \,x$$and$$y = {2^x}.$$Find the area of this region. Evaluate$$\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx} $$If$$'f$$is a continuous function with$$\int\limits_0^x {f\left( t \right)dt \to \infty } $$as$$\left| x \right| \to \infty ,$$then show that eve... Prove that for any positive integer$$k$$,$${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \rig...
Compute the area of the region bounded by the curves $$\,y = ex\,\ln x$$ and $$y = {{\ln x} \over {ex}}$$ where $$ln$$ $$e=1.$$
Show that $$\int\limits_0^{\pi /2} {f\left( {\sin 2x} \right)\sin x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\cos 2x} \right)\cos x\,dx}$$
If $$f$$ and $$g$$ are continuous function on $$\left[ {0,a} \right]$$ satisfying $$f\left( x \right) = f\left( {a - x} \right)$$ and $$g\left( x \ri... Find the area of the region bounded by the curve$$C:y=\tan x,$$tangent drawn to$$C$$at$$x = {\pi \over 4}$$and the$$x$$-axis. Evaluate$$\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx} $$Find the area bounded by the curves,$${x^2} + {y^2} = 25,\,4y = \left| {4 - {x^2}} \right|$$and$$x=0$$above the$$x$$-axis. Evaluate :$$\int\limits_0^\pi {{{x\,dx} \over {1 + \cos \,\alpha \,\sin x}},0 < \alpha < \pi } $$Sketch the region bounded by the curves$$y = \sqrt {5 - {x^2}} $$and$$y = \left| {x - 1} \right|$$and find its area. Evaluate the following :$$\,\,\int\limits_0^{\pi /2} {{{x\sin x\cos x} \over {{{\cos }^4}x + {{\sin }^4}x}}} dx$$Evaluate the following$$\int\limits_0^{{1 \over 2}} {{{x{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx} $$Given a function$$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$$and a real ... Find the area of the region bounded by the$$x$$-axis and the curves defined by$$$y = \tan x, - {\pi \over 3} \le x \le {\pi \over 3};\,\,y = \cot... Evaluate : $$\int\limits_0^{\pi /4} {{{\sin x + \cos x} \over {9 + 16\sin 2x}}dx}$$ 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... Find the value of $$\int\limits_{ - 1}^{3/2} {\left| {x\sin \,\pi \,x} \right|\,dx}$$ Show that $$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} = {\pi \over 2}\int\limits_0^\pi {f\left( {\sin x} \right)dx.}$$ 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... Find the area bounded by the curve $${x^2} = 4y$$ and the straight 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 The value of $$\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx}$$ is ............... Let $${d \over {dx}}\,F\left( x \right) = {{{e^{\sin x}}} \over x},\,x > 0.$$ If $$\int_1^4 {{{2{e^{\sin {x^2}}}} \over x}} \,\,dx = F\left( k \rig... If for nonzero$$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} = ...... For $$n>0,$$ $$\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx =$$ The value of $$\int\limits_2^3 {{{\sqrt x } \over {\sqrt {3 - x} + \sqrt x }}} dx$$ is ........... The value of $$\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi }$$ is .............. The value of $$\int\limits_{ - 2}^2 {\left| {1 - {x^2}} \right|dx}$$ is ............... The integral $$\int\limits_0^{1.5} {\left[ {{x^2}} \right]dx,}$$ Where [ ] denotes the greatest integer function, equals ............. $$f\left( x \right) = \left| {\matrix{ {\sec x} & {\cos x} & {{{\sec }^2}x + \cot x\cos ec\,x} \cr {{{\cos }^2}x} & {{{\cos }^2}x}... ## True or False The value of the integral$$\int\limits_0^{2a} {[{{f\left( x \right)} \over {\left\{ {f\left( x \right) + f\left( {2a - x} \right)} \right\}}}]\,dx}$...
EXAM MAP
Joint Entrance Examination