JEE Main
Mathematics
Definite Integrals and Applications of Integrals
Previous Years Questions

The value of the integral $$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{x + {\pi \over 4}} \over {2 - \cos 2x}}dx}$$ is :
The area of the region given by $$\{ (x,y):xy \le 8,1 \le y \le {x^2}\}$$ is :
$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} \right]$$ i...
Let $\alpha>0$. If $\int\limits_0^\alpha \frac{x}{\sqrt{x+\alpha}-\sqrt{x}} \mathrm{~d} x=\frac{16+20 \sqrt{2}}{15}$, then $\alpha$ is equal to :
If $\phi(x)=\frac{1}{\sqrt{x}} \int\limits_{\frac{\pi}{4}}^x\left(4 \sqrt{2} \sin t-3 \phi^{\prime}(t)\right) d t, x>0$, then \emptyset^{\prime}\lef... Let a differentiable function $$f$$ satisfy $$f(x)+\int_\limits{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3$$. Then $$12 f(8)$$ is equal to :... Let $$\alpha \in (0,1)$$ and $$\beta = {\log _e}(1 - \alpha )$$. Let $${P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}}... The value of$$\int_\limits{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x$$is equal to Let q be the maximum integral value of p in [0,10] for which the roots of the equation x^2-p x+\frac{5}{4} p=0 are rational. Then the area of ... If [t] denotes the greatest integer$$\le \mathrm{t}$$, then the value of$${{3(e - 1)} \over e}\int\limits_1^2 {{x^2}{e^{[x] + [{x^3}]}}dx} $$is :... The area of the region$$A = \left\{ {(x,y):\left| {\cos x - \sin x} \right| \le y \le \sin x,0 \le x \le {\pi \over 2}} \right\}$$is The value of the integral$$\int_1^2 {\left( {{{{t^4} + 1} \over {{t^6} + 1}}} \right)dt} $$is The value of the integral$$\int\limits_{1/2}^2 {{{{{\tan }^{ - 1}}x} \over x}dx} $$is equal to Let$$f(x) = x + {a \over {{\pi ^2} - 4}}\sin x + {b \over {{\pi ^2} - 4}}\cos x,x \in R$$be a function which satisfies$$f(x) = x + \int\limits_0^{\... Let $$\Delta$$ be the area of the region $$\left\{ {(x,y) \in {R^2}:{x^2} + {y^2} \le 21,{y^2} \le 4x,x \ge 1} \right\}$$. Then $${1 \over 2}\left( {\... Let$$[x]$$denote the greatest integer$$\le x$$. Consider the function$$f(x) = \max \left\{ {{x^2},1 + [x]} \right\}$$. Then the value of the integ... Let$$A=\left\{(x, y) \in \mathbb{R}^{2}: y \geq 0,2 x \leq y \leq \sqrt{4-(x-1)^{2}}\right\}$$and$$ B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R... The integral $$16\int\limits_1^2 {{{dx} \over {{x^3}{{\left( {{x^2} + 2} \right)}^2}}}}$$ is equal to The minimum value of the function $$f(x) = \int\limits_0^2 {{e^{|x - t|}}dt}$$ is : $$\int_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}} {{{48} \over {\sqrt {9 - 4{x^2}} }}dx}$$ is equal to The area enclosed by the curves $${y^2} + 4x = 4$$ and $$y - 2x = 2$$ is : If $$[t]$$ denotes the greatest integer $$\leq t$$, then the value of $$\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|+1\right] \mathrm{d} x$$ is :... The integral $$\int\limits_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} \mathrm{~d} x$$ is equal to : If $$f(\alpha)=\int\limits_{1}^{\alpha} \frac{\log _{10} \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}, \alpha>0$$, then $$f\left(\mathrm{e}^{3}\right)+f\left... The area of the region$$\left\{(x, y):|x-1| \leq y \leq \sqrt{5-x^{2}}\right\}$$is equal to : Let$$I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3, \ldots .$$Then : The area enclosed by the curves$$y=\log _{e}\left(x+\mathrm{e}^{2}\right), x=\log _{e}\left(\frac{2}{y}\right)$$and$$x=\log _{\mathrm{e}} 2$$, abov... Let$$f(x)=2+|x|-|x-1|+|x+1|, x \in \mathbf{R}$$. Consider$$(\mathrm{S} 1): f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f... The area of the region enclosed by $$y \leq 4 x^{2}, x^{2} \leq 9 y$$ and $$y \leq 4$$, is equal to : $$\int\limits_{0}^{2}\left(\left|2 x^{2}-3 x\right|+\left[x-\frac{1}{2}\right]\right) \mathrm{d} x$$, where [t] is the greatest integer function, is e... Consider a curve $$y=y(x)$$ in the first quadrant as shown in the figure. Let the area $$\mathrm{A}_{1}$$ is twice the area $$\mathrm{A}_{2}$$. Then t... Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a function defined as $$f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x], a \in \mathbb{R}$$ where $$[t... Let$$ I=\int_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x $$. Then The area of the smaller region enclosed by the curves$$y^{2}=8 x+4$$and$$x^{2}+y^{2}+4 \sqrt{3} x-4=0$$is equal to Let a function$$f: \mathbb{R} \rightarrow \mathbb{R}$$be defined as :$$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , ... $$\int\limits_{0}^{20 \pi}(|\sin x|+|\cos x|)^{2} d x \text { is equal to }$$ The area bounded by the curves $$y=\left|x^{2}-1\right|$$ and $$y=1$$ is The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = ya is $${{364} \over 3}$$, is equal to :... If $$a = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{{2n} \over {{n^2} + {k^2}}}}$$ and $$f(x) = \sqrt {{{1 - \cos x} \over {1 + \...$$\mathop {\lim }\limits_{n \to \infty } {1 \over {{2^n}}}\left( {{1 \over {\sqrt {1 - {1 \over {{2^n}}}} }} + {1 \over {\sqrt {1 - {2 \over {{2^n}}}}... Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of the integral $$\int_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 ... Let a smooth curve$$y=f(x)$$be such that the slope of the tangent at any point$$(x, y)$$on it is directly proportional to$$\left(\frac{-y}{x}\rig... The area of the region given by $$A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}$$ is : For any real number $$x$$, let $$[x]$$ denote the largest integer less than equal to $$x$$. Let $$f$$ be a real valued function defined on the interva... The slope of the tangent to a curve $$C: y=y(x)$$ at any point $$(x, y)$$ on it is $$\frac{2 \mathrm{e}^{2 x}-6 \mathrm{e}^{-x}+9}{2+9 \mathrm{e}^{-2 ... Let the locus of the centre$$(\alpha, \beta), \beta>0$$, of the circle which touches the circle$$x^{2}+(y-1)^{2}=1$$externally and also touches the...$$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}} $$is equal to : Let$${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R$$, represents a circle with center ($$\alpha$$,$$\beta$$). Then,$$\alpha... Let f be a real valued continuous function on [0, 1] and $$f(x) = x + \int\limits_0^1 {(x - t)f(t)dt}$$. Then, which of the following points (x, y) l... If $$\int\limits_0^2 {\left( {\sqrt {2x} - \sqrt {2x - {x^2}} } \right)dx = \int\limits_0^1 {\left( {1 - \sqrt {1 - {y^2}} - {{{y^2}} \over 2}} \rig... Let$$f:R \to R$$be a function defined by :$$f(x) = \left\{ {\matrix{ {\max \,\{ {t^3} - 3t\} \,t \le x} & ; & {x \le 2} \cr {{x^2} + 2x - 6... The area enclosed by y2 = 8x and y = $$\sqrt2$$ x that lies outside the triangle formed by y = $$\sqrt2$$ x, x = 1, y = 2$$\sqrt2$$, is equal to:... $$\int_0^5 {\cos \left( {\pi \left( {x - \left[ {{x \over 2}} \right]} \right)} \right)dx}$$, where [t] denotes greatest integer less than or equal t... Let f : R $$\to$$ R be a differentiable function such that $$f\left( {{\pi \over 4}} \right) = \sqrt 2 ,\,f\left( {{\pi \over 2}} \right) = 0$$ and ... Let f : R $$\to$$ R be a continuous function satisfying f(x) + f(x + k) = n, for all x $$\in$$ R where k > 0 and n is a positive integer. If $${I_1} ... The area of the bounded region enclosed by the curve$$y = 3 - \left| {x - {1 \over 2}} \right| - |x + 1|$$and the x-axis is : Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2$$\tan x(\cos x - y)$$. If the curve passes through the point$$\left( {{\pi ... Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral $$\int\limits_0^1 {[ - 8{x^2} + 6x - 1]dx}$$ is equal to... The area of the region S = {(x, y) : y2 $$\le$$ 8x, y $$\ge$$ $$\sqrt2$$x, x $$\ge$$ 1} is If m and n respectively are the number of local maximum and local minimum points of the function $$f(x) = \int\limits_0^{{x^2}} {{{{t^2} - 5t + 4} \ov... Let f be a differentiable function in$$\left( {0,{\pi \over 2}} \right)$$. If$$\int\limits_{\cos x}^1 {{t^2}\,f(t)dt = {{\sin }^3}x + \cos x} $$, t... The integral$$\int\limits_0^1 {{1 \over {{7^{\left[ {{1 \over x}} \right]}}}}dx} $$, where [ . ] denotes the greatest integer function, is equal to... The value of the integral$$\int\limits_{ - 2}^2 {{{|{x^3} + x|} \over {({e^{x|x|}} + 1)}}dx} $$is equal to : The area of the region bounded by y2 = 8x and y2 = 16(3$$-$$x) is equal to: The area bounded by the curve y = |x2$$-$$9| and the line y = 3 is : The area of the region enclosed between the parabolas y2 = 2x$$-$$1 and y2 = 4x$$-$$3 is The value of$$\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx} $$is equal to: The value of the integral$$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}} $$is equal to$$\mathop {\lim }\limits_{n \to \infty } \left( {{{{n^2}} \over {({n^2} + 1)(n + 1)}} + {{{n^2}} \over {({n^2} + 4)(n + 2)}} + {{{n^2}} \over {({n^2} ... Let f : R $$\to$$ R be a continuous function. Then $$\mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\limits_2^{{{\sec }^2}x} {f(x)\... The area, enclosed by the curves$$y = \sin x + \cos x$$and$$y = \left| {\cos x - \sin x} \right|$$and the lines$$x = 0,x = {\pi \over 2}$$, is : The function f(x), that satisfies the condition$$f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy} $$, is : If [x] is the greatest integer$$\le$$x, then$${\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x]} {)^{[x]}}dx$$is equal to :... Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If$$\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} } $$, ... The area of the region bounded by the parabola (y$$-$$2)2 = (x$$-$$1), the tangent to it at the point whose ordinate is 3 and the x-axis is :... The value of the integral$$\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}} $$is : If$${U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2}} \over {{n^2}}}} \right)^n$$...$$\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx} $$is equal to : If the value of the integral$$\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta } $$, where$$\alpha$$,$$\beta\i... The value of $$\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}}} \right)} \,dx$$ is The value of $$\int\limits_{{{ - 1} \over {\sqrt 2 }}}^{{1 \over {\sqrt 2 }}} {{{\left( {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{... The value of$$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{r = 0}^{2n - 1} {{{{n^2}} \over {{n^2} + 4{r^2}}}} $$is : The area of the region bounded by y$$-$$x = 2 and x2 = y is equal to : Let f : (a, b)$$\to$$R be twice differentiable function such that$$f(x) = \int_a^x {g(t)dt} $$for a differentiable function g(x). If f(x) = 0 has ... The value of$$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{j = 1}^n {{{(2j - 1) + 8n} \over {(2j - 1) + 4n}}} $$is equal to : The value of the definite integral$$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}}... If the area of the bounded region $$R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}$$ is , $$\alpha {({... If$$f(x) = \left\{ {\matrix{ {\int\limits_0^x {\left( {5 + \left| {1 - t} \right|} \right)dt,} } & {x > 2} \cr {5x + 1,} & {x \le ... The value of the integral $$\int\limits_{ - 1}^1 {\log \left( {x + \sqrt {{x^2} + 1} } \right)dx}$$ is : The value of the definite integral $$\int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}}$$ is : The area (in sq. units) of the region, given by the set $$\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\}$$ is : Let $$f:[0,\infty ) \to [0,\infty )$$ be defined as $$f(x) = \int_0^x {[y]dy}$$where [x] is the greatest integer less than or equal to x. Which of th... If $$\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} ... If [x] denotes the greatest integer less than or equal to x, then the value of the integral$$\int_{ - \pi /2}^{\pi /2} {[[x] - \sin x]dx} $$is equal... If the real part of the complex number$${(1 - \cos \theta + 2i\sin \theta )^{ - 1}}$$is$${1 \over 5}$$for$$\theta \in (0,\pi )$$, then the valu... Let y = y(x) satisfies the equation$${{dy} \over {dx}} - |A| = 0$$, for all x > 0, where$$A = \left[ {\matrix{ y & {\sin x} & 1 \cr ... Let $$g(t) = \int_{ - \pi /2}^{\pi /2} {\cos \left( {{\pi \over 4}t + f(x)} \right)} dx$$, where $$f(x) = {\log _e}\left( {x + \sqrt {{x^2} + 1} } \r... Let a be a positive real number such that$$\int_0^a {{e^{x - [x]}}} dx = 10e - 9$$where [ x ] is the greatest integer less than or equal to x. Then ... The value of the integral$$\int\limits_{ - 1}^1 {{{\log }_e}(\sqrt {1 - x} + \sqrt {1 + x} )dx} $$is equal to: The area bounded by the curve 4y2 = x2(4$$-$$x)(x$$-$$2) is equal to : Let g(x) =$$\int_0^x {f(t)dt} $$, where f is continuous function in [ 0, 3 ] such that$${1 \over 3} \le $$f(t)$$ \le $$1 for all t$$\in$$[0... Let f : R$$ \to $$R be defined as f(x) = e$$-$$xsinx. If F : [0, 1]$$ \to $$R is a differentiable function with that F(x) =$$\int_0^x {f(t)dt} $$... If the integral$$\int_0^{10} {{{[\sin 2\pi x]} \over {{e^{x - [x]}}}}} dx = \alpha {e^{ - 1}} + \beta {e^{ - {1 \over 2}}} + \gamma $$, where$$\alp... Which of the following statements is correct for the function g($$\alpha$$) for $$\alpha$$ $$\in$$ R such that $$g(\alpha ) = \int\limits_{{\pi \over... Consider the integral$$I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx} $$, where [x] denotes the greatest integer less than or equal to x. T... Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that$$\int_0^1 {P(x)dx} $$= 1 and P(x) leaves remainder 5 when it is di... For x > 0, if$$f(x) = \int\limits_1^x {{{{{\log }_e}t} \over {(1 + t)}}dt} $$, then$$f(e) + f\left( {{1 \over e}} \right)$$is equal to : The value of$$\int\limits_{ - \pi /2}^{\pi /2} {{{{{\cos }^2}x} \over {1 + {3^x}}}} dx$$is : The value of$$\sum\limits_{n = 1}^{100} {\int\limits_{n - 1}^n {{e^{x - [x]}}dx} } $$, where [ x ] is the greatest integer$$ \le $$x, is : If$${I_n} = \int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\cot }^n}x\,dx} $$, then :$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over n} + {n \over {{{(n + 1)}^2}}} + {n \over {{{(n + 2)}^2}}} + ........ + {n \over {{{(2n + 1)... The value of $$\int\limits_{ - 1}^1 {{x^2}{e^{[{x^3}]}}} dx$$, where [ t ] denotes the greatest integer $$\le$$ t, is : If a curve y = f(x) passes through the point (1, 2) and satisfies $$x {{dy} \over {dx}} + y = b{x^4}$$, then for what value of b, $$\int\limits_1^2 {f... The value of the integral,$$\int\limits_1^3 {[{x^2} - 2x - 2]dx} $$, where [x] denotes the greatest integer less than or equal to x, is : Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2$$-$$x) for all x$$ \in $$(0, 2), f(0) = 1 and f(2) = e2. Then the va... The area of the region :$$R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\} $$is :$$\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {\left( {\sin \sqrt t } \right)dt} } \over {{x^3}}}$$is equal to : The area (in sq. units) of the part of the circle x2 + y2 = 36, which is outside the parabola y2 = 9x, is : The integral$$\int\limits_1^2 {{e^x}.{x^x}\left( {2 + {{\log }_e}x} \right)} dx$$equals : The area (in sq. units) of the region enclosed by the curves y = x2 – 1 and y = 1 – x2 is equal to : The area (in sq. units) of the region A = {(x, y) : |x| + |y|$$ \le $$1, 2y2$$ \ge $$|x|} If I1 =$$\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx$$and I2 =$$\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx$$suc... The area (in sq. units) of the region A = {(x, y) : (x – 1)[x]$$ \le $$y$$ \le $$2$$\sqrt x $$, 0$$ \le $$x$$ \le $$2}, where [t] denotes the... The value of$$\int\limits_{{{ - \pi } \over 2}}^{{\pi \over 2}} {{1 \over {1 + {e^{\sin x}}}}dx} $$is: The integral$$\int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{\tan }^3}x.{{\sin }^2}3x\left( {2{{\sec }^2}x.{{\sin }^2}3x + 3\tan x.\sin 6x} \right)... Let $$f\left( x \right) = \int {{{\sqrt x } \over {{{\left( {1 + x} \right)}^2}}}dx\left( {x \ge 0} \right)}$$. Then f(3) – f(1) is eqaul to : Let $$f(x) = \left| {x - 2} \right|$$ and g(x) = f(f(x)), $$x \in \left[ {0,4} \right]$$. Then $$\int\limits_0^3 {\left( {g(x) - f(x)} \right)} dx$$ i... If the value of the integral $$\int\limits_0^{{1 \over 2}} {{{{x^2}} \over {{{\left( {1 - {x^2}} \right)}^{{3 \over 2}}}}}} dx$$ is $${k \over 6}$$,... Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If T = {x $$\in$$ R | f(x) = f(0)}, then the sum of squares of all... If x3dy + xy dx = x2dy + 2y dx; y(2) = e and x > 1, then y(4) is equal to : The area (in sq. units) of the region { (x, y) : 0 $$\le$$ y $$\le$$ x2 + 1, 0 $$\le$$ y $$\le$$ x + 1, $${1 \over 2}$$ $$\le$$ x $$\le$$... $$\int\limits_{ - \pi }^\pi {\left| {\pi - \left| x \right|} \right|dx}$$ is equal to : Consider a region R = {(x, y) $$\in$$ R : x2 $$\le$$ y $$\le$$ 2x}. if a line y = $$\alpha$$ divides the area of region R into two equal parts,... Area (in sq. units) of the region outside $${{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1$$ and inside the ellipse $${{{x^2}} \over ... Let y = y(x) be the solution of the differential equation,$${{2 + \sin x} \over {y + 1}}.{{dy} \over {dx}} = - \cos x$$, y > 0,y(0) = 1. If y($$\... Given : $$f(x) = \left\{ {\matrix{ {x\,\,\,\,\,,} & {0 \le x < {1 \over 2}} \cr {{1 \over 2}\,\,\,\,,} & {x = {1 \over 2}} \cr ... The value of$$\int\limits_0^{2\pi } {{{x{{\sin }^8}x} \over {{{\sin }^8}x + {{\cos }^8}x}}} dx$$is equal to : If for all real triplets (a, b, c), ƒ(x) = a + bx + cx2; then$$\int\limits_0^1 {f(x)dx} $$is equal to : The area (in sq. units) of the region {(x,y)$$ \in $$R2 : x2$$ \le $$y$$ \le $$3 – 2x}, is If$$I = \int\limits_1^2 {{{dx} \over {\sqrt {2{x^3} - 9{x^2} + 12x + 4} }}} $$, then : Let ƒ(x) = (sin(tan–1x) + sin(cot–1x))2 – 1, |x| > 1. If$${{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\sin }^{ - 1}}\left( {f\left( x \... For a > 0, let the curves C1 : y2 = ax and C2 : x2 = ay intersect at origin O and a point P. Let the line x = b (0 < b < a) intersect the cho... The value of $$\alpha$$ for which $$4\alpha \int\limits_{ - 1}^2 {{e^{ - \alpha \left| x \right|}}dx} = 5$$, is: The area (in sq. units) of the region {(x, y) $$\in$$ R2 | 4x2 $$\le$$ y $$\le$$ 8x + 12} is : If $$\theta$$1 and $$\theta$$2 be respectively the smallest and the largest values of $$\theta$$ in (0, 2$$\pi$$) - {$$\pi$$} which satisfy the... The area of the region, enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = ... If ƒ(a + b + 1 - x) = ƒ(x), for all x, where a and b are fixed positive real numbers, then $${1 \over {a + b}}\int_a^b {x\left( {f(x) + f(x + 1)} \rig... If the area (in sq. units) bounded by the parabola y2 = 4$$\lambda $$x and the line y =$$\lambda $$x,$$\lambda $$> 0, is$${1 \over 9}$$, then... A value of$$\alpha $$such that$$\int\limits_\alpha ^{\alpha + 1} {{{dx} \over {\left( {x + \alpha } \right)\left( {x + \alpha + 1} \right)}}} = ... If $$\int\limits_0^{{\pi \over 2}} {{{\cot x} \over {\cot x + \cos ecx}}} dx$$ = m($$\pi$$ + n), then m.n is equal to If the area (in sq. units) of the region {(x, y) : y2 $$\le$$ 4x, x + y $$\le$$ 1, x $$\ge$$ 0, y $$\ge$$ 0} is a $$\sqrt 2$$ + b, then a – b... Let f : R $$\to$$ R be a continuously differentiable function such that f(2) = 6 and f'(2) = $${1 \over {48}}$$. If $$\int\limits_6^{f\left( x \righ... The area (in sq.units) of the region bounded by the curves y = 2x and y = |x + 1|, in the first quadrant is : The integral$$\int\limits_{\pi /6}^{\pi /3} {{{\sec }^{2/3}}} x\cos e{c^{4/3}}xdx$$is equal to : The value of$$\int\limits_0^{2\pi } {\left[ {\sin 2x\left( {1 + \cos 3x} \right)} \right]} dx$$, where [t] denotes the greatest integer function is :... The value of the integral$$\int\limits_0^1 {x{{\cot }^{ - 1}}(1 - {x^2} + {x^4})dx} $$is :- If f : R$$ \to $$R is a differentiable function and f(2) = 6, then$$\mathop {\lim }\limits_{x \to 2} {{\int\limits_6^{f\left( x \right)} {2tdt} } \... The area (in sq. units) of the region A = {(x, y) : $${{y{}^2} \over 2}$$ $$\le$$ x $$\le$$ y + 4} is :- The value of $$\int\limits_0^{\pi /2} {{{{{\sin }^3}x} \over {\sin x + \cos x}}dx}$$ is The area (in sq. units) of the region A = {(x, y) : x2 $$\le$$ y $$\le$$ x + 2} is Let S($$\alpha$$) = {(x, y) : y2 $$\le$$ x, 0 $$\le$$ x $$\le$$ $$\alpha$$} and A($$\alpha$$) is area of the region S($$\alpha$$). If for a... If $$f(x) = {{2 - x\cos x} \over {2 + x\cos x}}$$ and g(x) = logex, (x > 0) then the value of integral $$\int\limits_{ - {\pi \over 4}}^{{\pi \ov... The area (in sq. units) of the region A = { (x, y)$$ \in $$R × R| 0$$ \le $$x$$ \le $$3, 0$$ \le $$y$$ \le $$4, y$$ \le $$x2 ... The integral$$\int\limits_1^e {\left\{ {{{\left( {{x \over e}} \right)}^{2x}} - {{\left( {{e \over x}} \right)}^x}} \right\}} \,$$loge x dx is equal... The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is The area (in sq. units) in the first quadrant bounded by the parabola, y = x2 + 1, the tangent to it at the point (2, 5) and the coordinate axes is :... The integral$$\int\limits_{\pi /6}^{\pi /4} {{{dx} \over {\sin 2x\left( {{{\tan }^5}x + {{\cot }^5}x} \right)}}} $$equals : The value of the integral$$\int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \,dx$$(where [x] denotes ... If$$\int {{{\sqrt {1 - {x^2}} } \over {{x^4}}}} $$dx = A(x)$${\left( {\sqrt {1 - {x^2}} } \right)^m}$$+ C, for a suitable chosen integer... The area (in sq. units) of the region bounded by the curve x2 = 4y and the straight line x = 4y – 2 is : The value of$$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {\left[ x \right] + \left[ {\sin x} \right] + 4}}} ,$$where [t] d... If$$\int\limits_0^x \, $$f(t) dt = x2 +$$\int\limits_x^1 \, $$t2f(t) dt then f '$$\left( {{1 \over 2}} \right)$$is -... Let$${\rm I} = \int\limits_a^b {\left( {{x^4} - 2{x^2}} \right)} dx.$$If I is minimum then the ordered pair (a, b) is - If the area enclosed between the curves y = kx2 and x = ky2, (k > 0), is 1 square unit. Then k is - The area of the region A = {(x, y) : 0$$ \le $$y$$ \le $$x |x| + 1 and$$-$$1$$ \le $$x$$ \le $$1} in sq. units, is : ... If$$\int\limits_0^{{\pi \over 3}} {{{\tan \theta } \over {\sqrt {2k\,\sec \theta } }}} \,d\theta = 1 - {1 \over {\sqrt 2 }},\left(... The value of $$\int\limits_0^\pi {{{\left| {\cos x} \right|}^3}} \,dx$$ is : The area (in sq. units) bounded by the parabolae y = x2 – 1, the tangent at the point (2, 3) to it and the y-axis is : If $$f(x) = \int\limits_0^x {t\left( {\sin x - \sin t} \right)dt\,\,\,}$$ then : If the area of the region bounded by the curves, $$y = {x^2},y = {1 \over x}$$ and the lines y = 0 and x= t (t >1) is 1 sq. unit, then t is equal... Let g(x) = cosx2, f(x) = $$\sqrt x$$ and $$\alpha ,\beta \left( {\alpha < \beta } \right)$$ be the roots of the quadratic equation 18x2 - 9$$\pi ... The value of$$\int\limits_{ - \pi /2}^{\pi /2} {{{{{\sin }^2}x} \over {1 + {2^x}}}} dx$$is The value of integral$$\int_{{\pi \over 4}}^{{{3\pi } \over 4}} {{x \over {1 + \sin x}}dx} $$is : If$${I_1} = \int_0^1 {{e^{ - x}}} {\cos ^2}x{\mkern 1mu} dx;{I_2} = \int_0^1 {{e^{ - {x^2}}}} {\cos ^2}x{\mkern 1mu} d... The area (in sq. units) of the region {x $$\in$$ R : x $$\ge$$ 0, y $$\ge$$ 0, y $$\ge$$ x $$-$$ 2 and y $$\le$$ $$\sqrt x$$}, is :... The value of the integral $$\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {{{\sin }^4}} x\left( {1 + \log \left( {{{2 + \sin x} \over {2 - \sin x}}... If$$\int\limits_1^2 {{{dx} \over {{{\left( {{x^2} - 2x + 4} \right)}^{{3 \over 2}}}}}} = {k \over {k + 5}},$$then k is equal to : If$$\mathop {\lim }\limits_{n \to \infty } \,\,{{{1^a} + {2^a} + ...... + {n^a}} \over {{{(n + 1)}^{a - 1}}\left[ {\left( {na + 1} \righ... The area (in sq. units) of the smaller portion enclosed between the curves, x2 + y2 = 4 and y2 = 3x, is : The integral $$\int_{{\pi \over {12}}}^{{\pi \over 4}} {\,\,{{8\cos 2x} \over {{{\left( {\tan x + \cot x} \right)}^3}}}} \,dx$$ equals : The integral $$\int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{dx} \over {1 + \cos x}}}$$ is equal to The area (in sq. units) of the region $$\left\{ {\left( {x,y} \right):x \ge 0,x + y \le 3,{x^2} \le 4y\,and\,y \le 1 + \sqrt x } \right\}$$ is The value of the integral $$\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} ,$$ ... For x $$\in$$ R, x $$\ne$$ 0, if y(x) is a differentiable function such that x $$\int\limits_1^x y$$ (t) dt = (x + 1) $$\int\limits_1^x ty$$ (t... If $$2\int\limits_0^1 {{{\tan }^{ - 1}}xdx = \int\limits_0^1 {{{\cot }^{ - 1}}} } \left( {1 - x + {x^2}} \right)dx,$$ then $$\int\limits_0... The area (in sq. units) of the region described by A= {(x, y)$$\left| {} \right.$$y$$ \ge $$x2$$-$$5x + 4, x + y$$ \ge $$1, y$$ \le $$0} is : ... The area (in sq. units) of the region$$\left\{ {\left( {x,y} \right):{y^2} \ge 2x\,\,\,and\,\,\,{x^2} + {y^2} \le 4x,x \ge 0,y \ge 0} \right\}$$is : The integral$$\int\limits_2^4 {{{\log \,{x^2}} \over {\log {x^2} + \log \left( {36 - 12x + {x^2}} \right)}}dx} $$is equal to : The area (in sq. units) of the region described by$$\left\{ {\left( {x,y} \right):{y^2} \le 2x} \right.$$and$$\left. {y \ge 4x - 1} \right\}$$is ... The area of the region described by$$A = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right.$$and$$\left. {{y^2} \le 1 - x} \right\}$$is :... The integral$$\int\limits_0^\pi {\sqrt {1 + 4{{\sin }^2}{x \over 2} - 4\sin {x \over 2}{\mkern 1mu} } } dx$$equals: Statement-1 : The value of the integral$$\int\limits_{\pi /6}^{\pi /3} {{{dx} \over {1 + \sqrt {\tan \,x} }}} $$is equal to$$\pi /6$$Statement-2 ... The area (in square units) bounded by the curves$$y = \sqrt {x,} 2y - x + 3 = 0,x$$-axis, and lying in the first quadrant is : The area between the parabolas$${x^2} = {y \over 4}$$and$${x^2} = 9y$$and the straight line$$y=2$$is : The value of$$\int\limits_0^1 {{{8\log \left( {1 + x} \right)} \over {1 + {x^2}}}} dx$$is The area of the region enclosed by the curves$$y = x,x = e,y = {1 \over x}$$and the positive$$x$$-axis is The area bounded by the curves$$y = \cos x$$and$$y = \sin x$$between the ordinates$$x=0$$and$$x = {{3\pi } \over 2}$$is Let$$p(x)$$be a function defined on$$R$$such that$$p'(x)=p'(1-x),$$for all$$x \in \left[ {0,1} \right],p\left( 0 \right) = 1$$and$$p(1)=41.$$... The area of the region bounded by the parabola$${\left( {y - 2} \right)^2} = x - 1,$$the tangent of the parabola at the point$$(2, 3)$$and the$$x... $$\int\limits_0^\pi {\left[ {\cot x} \right]dx,}$$ where $$\left[ . \right]$$ denotes the greatest integer function, is equal to: The area of the plane region bounded by the curves $$x + 2{y^2} = 0$$ and $$\,x + 3{y^2} = 1$$ is equal to Let $$F\left( x \right) = f\left( x \right) + f\left( {{1 \over x}} \right),$$ where $$f\left( x \right) = \int\limits_l^x {{{\log t} \over {1 + t}}dt... The solution for$$x$$of the equation$$\int\limits_{\sqrt 2 }^x {{{dt} \over {t\sqrt {{t^2} - 1} }} = {\pi \over 2}} $$is The area enclosed between the curves$${y^2} = x$$and$$y = \left| x \right|$$is Let$$I = \int\limits_0^1 {{{\sin x} \over {\sqrt x }}dx} $$and$$J = \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx} .$$Then which one of the follo...$$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} $$is equal to$$\int\limits_{ - {{3\pi } \over 2}}^{ - {\pi \over 2}} {\left[ {{{\left( {x + \pi } \right)}^3} + {{\cos }^2}\left( {x + 3\pi } \right)} \right]} dx... The value of $$\int\limits_1^a {\left[ x \right]} f'\left( x \right)dx,a > 1$$ where $${\left[ x \right]}$$ denotes the greatest integer not exceed... If $${I_1} = \int\limits_0^1 {{2^{{x^2}}}dx,{I_2} = \int\limits_0^1 {{2^{{x^3}}}dx,\,{I_3} = \int\limits_1^2 {{2^{{x^2}}}dx} } }$$ and $${I_4} = \int... The area enclosed between the curve$$y = {\log _e}\left( {x + e} \right)$$and the coordinate axes is The parabolas$${y^2} = 4x$$and$${x^2} = 4y$$divide the square region bounded by the lines$$x=4,y=4$$and the coordinate axes. If$${S_1},{S_... Let $$f(x)$$ be a non - negative continuous function such that the area bounded by the curve $$y=f(x),$$ $$x$$-axis and the ordinates $$x = {\pi \ov... The value of$$\int\limits_{ - \pi }^\pi {{{{{\cos }^2}} \over {1 + {a^x}}}dx,\,\,a > 0,} $$is The value of integral,$$\int\limits_3^6 {{{\sqrt x } \over {\sqrt {9 - x} + \sqrt x }}} dx $$is$$\mathop {Lim}\limits_{n \to \infty } \sum\limits_{r = 1}^n {{1 \over n}{e^{{r \over n}}}} $$is The value of$$I = \int\limits_0^{\pi /2} {{{{{\left( {\sin x + \cos x} \right)}^2}} \over {\sqrt {1 + \sin 2x} }}dx} $$is The value of$$\int\limits_{ - 2}^3 {\left| {1 - {x^2}} \right|dx} $$is If$$\int\limits_0^\pi {xf\left( {\sin x} \right)dx = A\int\limits_0^{\pi /2} {f\left( {\sin x} \right)dx,} } $$then$$A$$is If$$f\left( x \right) = {{{e^x}} \over {1 + {e^x}}},{I_1} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {xg\left\{ {x\left( {1 - x} \rig... The area of the region bounded by the curves $$y = \left| {x - 2} \right|,x = 1,x = 3$$ and the $$x$$-axis is The area of the region bounded by the curves $$y = \left| {x - 1} \right|$$ and $$y = 3 - \left| x \right|$$ is Let $$f(x)$$ be a function satisfying $$f'(x)=f(x)$$ with $$f(0)=1$$ and $$g(x)$$ be a function that satisfies $$f\left( x \right) + g\left( x \right)... If$$f\left( {a + b - x} \right) = f\left( x \right)$$then$$\int\limits_a^b {xf\left( x \right)dx} $$is equal to The value of the integral$$I = \int\limits_0^1 {x{{\left( {1 - x} \right)}^n}dx} $$is$${I_n} = \int\limits_0^{\pi /4} {{{\tan }^n}x\,dx} $$then$$\,\mathop {\lim }\limits_{n \to \infty } \,n\left[ {{I_n} + {I_{n + 2}}} \right]$$equal...$$\int_0^{10\pi } {\left| {\sin x} \right|dx} $$is$$\int\limits_0^2 {\left[ {{x^2}} \right]dx} $$is$$\int_{ - \pi }^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} dx$$is If$$y=f(x)$$makes +$$ve$$intercept of$$2$$and$$0$$unit on$$x$$and$$y$$axes and encloses an area of$$3/4$$square unit with the axes then ... The area bounded by the curves$$y = \ln x,y = \ln \left| x \right|,y = \left| {\ln {\mkern 1mu} x} \right|$$and$$y = \left| {\ln \left| x \right|} ... ## Numerical If $$\int\limits_0^\pi {{{{5^{\cos x}}(1 + \cos x\cos 3x + {{\cos }^2}x + {{\cos }^3}x\cos 3x)dx} \over {1 + {5^{\cos x}}}} = {{k\pi } \over {16}}} ... If$$\int_\limits{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x=\frac{1}{l}(11)^{m / n}$$where$$l, m, n \in \mat... Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a differentiable function such that $$f^{\prime}(x)+f(x)=\int_\limits{0}^{2} f(t) d t$$. If $$f(0)=e^{... Let$$A$$be the area bounded by the curve$$y=x|x-3|$$, the$$x$$-axis and the ordinates$$x=-1$$and$$x=2$$. Then$$12 A$$is equal to ____________... Let \mathrm{S} be the set of all \mathrm{a} \in \mathrm{N} such that the area of the triangle formed by the tangent at the point \mathrm{P}(\math... Let the area of the region \left\{(x, y):|2 x-1| \leq y \leq\left|x^{2}-x\right|, 0 \leq x \leq 1\right\} be \mathrm{A}. Then (6 \mathrm{~A}+11)^... Let for$$x \in \mathbb{R}$$,$$ f(x)=\frac{x+|x|}{2} \text { and } g(x)=\left\{\begin{array}{cc} x, & x Then area bounded by the curve $$y=(f \circ g... Let A be the area of the region \left\{(x, y): y \geq x^2, y \geq(1-x)^2, y \leq 2 x(1-x)\right\}. Then 540 \mathrm{~A} is equal to...$$\lim_\limits{x \rightarrow 0} \frac{48}{x^{4}} \int_\limits{0}^{x} \frac{t^{3}}{t^{6}+1} \mathrm{~d} t$$is equal to ___________. Let$$\alpha$$be the area of the larger region bounded by the curve$$y^{2}=8 x$$and the lines$$y=x$$and$$x=2$$, which lies in the first quadrant... If$$\int\limits_{{1 \over 3}}^3 {|{{\log }_e}x|dx = {m \over n}{{\log }_e}\left( {{{{n^2}} \over e}} \right)} $$, where m and n are coprime natural n... If the area enclosed by the parabolas$$\mathrm{P_1:2y=5x^2}$$and$$\mathrm{P_2:x^2-y+6=0}$$is equal to the area enclosed by$$\mathrm{P_1}$$and$$... Let $$f$$ be $$a$$ differentiable function defined on $$\left[ {0,{\pi \over 2}} \right]$$ such that $$f(x) > 0$$ and $$f(x) + \int_0^x {f(t)\sqrt {1... If the area of the region bounded by the curves$$y^2-2y=-x,x+y=0$$is A, then 8 A is equal to __________ The value of$$12\int\limits_0^3 {\left| {{x^2} - 3x + 2} \right|dx} $$is ____________ The value of$${8 \over \pi }\int\limits_0^{{\pi \over 2}} {{{{{(\cos x)}^{2023}}} \over {{{(\sin x)}^{2023}} + {{(\cos x)}^{2023}}}}dx} $$is ______... The value of the integral$$\int\limits_{0}^{\frac{\pi}{2}} 60 \frac{\sin (6 x)}{\sin x} d x$$is equal to _________. If$$\int\limits_{0}^{\sqrt{3}} \frac{15 x^{3}}{\sqrt{1+x^{2}+\sqrt{\left(1+x^{2}\right)^{3}}}} \mathrm{~d} x=\alpha \sqrt{2}+\beta \sqrt{3}$$, where ... Let$$f(x)=\min \{[x-1],[x-2], \ldots,[x-10]\}$$where [t] denotes the greatest integer$$\leq \mathrm{t}$$. Then$$\int\limits_{0}^{10} f(x) \mathrm{... Let f be a differentiable function satisfying $$f(x)=\frac{2}{\sqrt{3}} \int\limits_{0}^{\sqrt{3}} f\left(\frac{\lambda^{2} x}{3}\right) \mathrm{d} \l... If$$\mathrm{n}(2 \mathrm{n}+1) \int_{0}^{1}\left(1-x^{\mathrm{n}}\right)^{2 \mathrm{n}} \mathrm{d} x=1177 \int_{0}^{1}\left(1-x^{\mathrm{n}}\right)^{... Let a curve $$y=y(x)$$ pass through the point $$(3,3)$$ and the area of the region under this curve, above the $$x$$-axis and between the abscissae 3 ... Let $${a_n} = \int_{ - 1}^n {\left( {1 + {x \over 2} + {{{x^2}} \over 3} + \,\,.....\,\, + \,\,{{{x^{n - 1}}} \over n}} \right)dx}$$ for every n $$\i... Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve$$4{x^3} - 3x{y^2} + 6{x^2} - 5xy - 8{y^2} + 9x + 14 = 0$$at the p...$$ \begin{aligned} &\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\ &=33 \cdot \lim _{n \righ... If for some $$\alpha$$ > 0, the area of the region $$\{ (x,y):|x + \alpha | \le y \le 2 - |x|\}$$ is equal to $${3 \over 2}$$, then the area of the r... Let $$f(t) = \int\limits_0^t {{e^{{x^3}}}\left( {{{{x^8}} \over {{{({x^6} + 2{x^3} + 2)}^2}}}} \right)dx}$$. If $$f(1) + f'(1) = \alpha e - {1 \over ... For real numbers a, b (a > b > 0), let Area$$\left\{ {(x,y):{x^2} + {y^2} \le {a^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \ge 1} \r... If the area of the region $$\left\{ {(x,y):{x^{{2 \over 3}}} + {y^{{2 \over 3}}} \le 1,\,x + y \ge 0,\,y \ge 0} \right\}$$ is A, then $${{256A} \over ... Let$${A_1} = \left\{ {(x,y):|x| \le {y^2},|x| + 2y \le 8} \right\}$$and$${A_2} = \left\{ {(x,y):|x| + |y| \le k} \right\}$$. If 27 (Area A1) = 5 (A... The integral$${{24} \over \pi }\int_0^{\sqrt 2 } {{{(2 - {x^2})dx} \over {(2 + {x^2})\sqrt {4 + {x^4}} }}} $$is equal to ____________. Let f(x) = max {|x + 1|, |x + 2|, ....., |x + 5|}. Then$$\int\limits_{ - 6}^0 {f(x)dx} $$is equal to __________. The value of the integral$${{48} \over {{\pi ^4}}}\int\limits_0^\pi {\left( {{{3\pi {x^2}} \over 2} - {x^3}} \right){{\sin x} \over {1 + {{\cos }^2}... The value of b > 3 for which $$12\int\limits_3^b {{1 \over {({x^2} - 1)({x^2} - 4)}}dx = {{\log }_e}\left( {{{49} \over {40}}} \right)}$$, is equal t... The area (in sq. units) of the region enclosed between the parabola y2 = 2x and the line x + y = 4 is __________. Let $$f(\theta ) = \sin \theta + \int\limits_{ - \pi /2}^{\pi /2} {(\sin \theta + t\cos \theta )f(t)dt}$$. Then the value of $$\left| {\int_0^{\pi ... Let$$\mathop {Max}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \alpha $$and$$\mathop {Min}\limits_{0\, \le x\, \le ... Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1, R2. If max {R1, R2} = R2, then... If the line y = mx bisects the area enclosed by the lines x = 0, y = 0, x = $${3 \over 2}$$ and the curve y = 1 + 4x $$-$$ x2, then 12 m is equal to _... Let [t] denote the greatest integer $$\le$$ t. Then the value of $$8.\int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx}$$ is ___________. If $$x\phi (x) = \int\limits_5^x {(3{t^2} - 2\phi '(t))dt}$$, x > $$-$$2, and $$\phi$$(0) = 4, then $$\phi$$(2) is __________. Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x3 $$-$$ 3x2 $$-$$ 12x. If A is the total area of th... The area of the region $$S = \{ (x,y):3{x^2} \le 4y \le 6x + 24\}$$ is ____________. If $$\int_0^\pi {({{\sin }^3}x){e^{ - {{\sin }^2}x}}dx = \alpha - {\beta \over e}\int_0^1 {\sqrt t {e^t}dt} }$$, then $$\alpha$$ + $$\beta$$ is eq... Let the domain of the function$$f(x) = {\log _4}\left( {{{\log }_5}\left( {{{\log }_3}(18x - {x^2} - 77)} \right)} \right)$$ be (a, b). Then the value... Let a curve y = f(x) pass through the point (2, (loge2)2) and have slope $${{2y} \over {x{{\log }_e}x}}$$ for all positive real value of x. Then the v... The area (in sq. units) of the region bounded by the curves x2 + 2y $$-$$ 1 = 0, y2 + 4x $$-$$ 4 = 0 and y2 $$-$$ 4x $$-$$ 4 = 0, in the upper half pl... Let T be the tangent to the ellipse E : x2 + 4y2 = 5 at the point P(1, 1). If the area of the region bounded by the tangent T, ellipse E, lines x = 1 ... Let P(x) be a real polynomial of degree 3 which vanishes at x = $$-$$3. Let P(x) have local minima at x = 1, local maxima at x = $$-$$1 and $$\int\lim... Let f(x) and g(x) be two functions satisfying f(x2) + g(4$$-$$x) = 4x3 and g(4$$-$$x) + g(x) = 0, then the value of$$\int\limits_{ - 4}^4 {f{{(x)... Let $${I_n} = \int_1^e {{x^{19}}{{(\log |x|)}^n}} dx$$, where n$$\in$$N. If (20)I10 = $$\alpha$$I9 + $$\beta$$I8, for natural numbers $$\alpha$$ and...
Let f : [$$-$$3, 1] $$\to$$ R be given as $$f(x) = \left\{ \matrix{ \min \,\{ (x + 6),{x^2}\}, - 3 \le x \le 0 \hfill \cr \max \,\{ \sqrt x ,{x... If [ . ] represents the greatest integer function, then the value of$$\left| {\int\limits_0^{\sqrt {{\pi \over 2}} } {\left[ {[{x^2}] - \cos x} \rig...
Let the curve y = y(x) be the solution of the differential equation, $${{dy} \over {dx}}$$ = 2(x + 1). If the numerical value of area bounded by the c...
Let f : R $$\to$$ R be a continuous function such that f(x) + f(x + 1) = 2, for all x$$\in$$R. If $${I_1} = \int\limits_0^8 {f(x)dx}$$ and $${I_2} ... If the normal to the curve y(x) =$$\int\limits_0^x {(2{t^2} - 15t + 10)dt} $$at a point (a, b) is parallel to the line x + 3y =$$-$$5, a > 1, th... Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3,$$-$$3) and (4,$$-$$2$$\sqrt 2 ...
If $${I_{m,n}} = \int\limits_0^1 {{x^{m - 1}}{{(1 - x)}^{n - 1}}dx}$$, for m, $$n \ge 1$$, and $$\int\limits_0^1 {{{{x^{m - 1}} + {x^{n - 1}}} \over ... The value of the integral$$\int\limits_0^\pi {|{{\sin }\,}2x|dx} $$is ___________. The area bounded by the lines y = || x$$-$$1 |$$-$$2 | is ___________. The value of$$\int\limits_{ - 2}^2 {|3{x^2} - 3x - 6|dx} $$is ___________. The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs... If$$\int\limits_{ - a}^a {\left( {\left| x \right| + \left| {x - 2} \right|} \right)} dx = 22$$, (a > 2) and [x] denotes the greatest integer$$ \...
Let {x} and [x] denote the fractional part of x and the greatest integer $$\le$$ x respectively of a real number x. If $$\int_0^n {\left\{ x \right\... Let [t] denote the greatest integer less than or equal to t. Then the value of$$\int\limits_1^2 {\left| {2x - \left[ {3x} \right]} \right|dx} $$is _... The integral$$\int\limits_0^2 {\left| {\left| {x - 1} \right| - x} \right|dx} $$is equal to______. Let ƒ(x) be a polynomial of degree 3 such that ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point at x = –1 and ƒ'(x) has a critical point at x = 1. The... ## MCQ (More than One Correct Answer) If$$g\left( x \right) = \int\limits_0^x {\cos 4t\,dt,} $$then$$g\left( {x + \pi } \right) equals
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