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## Definite Integrals and Applications of Integrals

### Mathematics

#### Previous Years Questions

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If $$[t]$$ denotes the greatest integer $$\leq t$$, then the value of $$\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|... JEE Main 2022 (Online) 29th July Evening Shift The area of the region$$\left\{(x, y):|x-1| \leq y \leq \sqrt{5-x^{2}}\right\}$$is equal to : JEE Main 2022 (Online) 29th July Morning Shift If$$f(\alpha)=\int\limits_{1}^{\alpha} \frac{\log _{10} \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}, \alpha>0$$, then$$f\lef...
JEE Main 2022 (Online) 29th July Morning Shift
The integral $$\int\limits_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} \mathrm{~d} x$$ is equal to :
JEE Main 2022 (Online) 29th July Morning Shift
The area enclosed by the curves $$y=\log _{e}\left(x+\mathrm{e}^{2}\right), x=\log _{e}\left(\frac{2}{y}\right)$$ and $$... JEE Main 2022 (Online) 28th July Evening Shift Let$$I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3, \ldots .$$Then : JEE Main 2022 (Online) 28th July Evening Shift Consider a curve$$y=y(x)$$in the first quadrant as shown in the figure. Let the area$$\mathrm{A}_{1}$$is twice the a... JEE Main 2022 (Online) 27th July Evening Shift$$\int\limits_{0}^{2}\left(\left|2 x^{2}-3 x\right|+\left[x-\frac{1}{2}\right]\right) \mathrm{d} x$$, where [t] is the g... JEE Main 2022 (Online) 27th July Evening Shift The area of the region enclosed by$$y \leq 4 x^{2}, x^{2} \leq 9 y$$and$$y \leq 4$$, is equal to : JEE Main 2022 (Online) 27th July Evening Shift Let$$f(x)=2+|x|-|x-1|+|x+1|, x \in \mathbf{R}$$. Consider$$(\mathrm{S} 1): f^{\prime}\left(-\frac{3}{2}\right)+f^{\pri...
JEE Main 2022 (Online) 27th July Evening Shift
Let a function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be defined as : $$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|... JEE Main 2022 (Online) 27th July Morning Shift The area of the smaller region enclosed by the curves$$y^{2}=8 x+4$$and$$x^{2}+y^{2}+4 \sqrt{3} x-4=0$$is equal to... JEE Main 2022 (Online) 27th July Morning Shift Let$$ I=\int_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x $$. Then JEE Main 2022 (Online) 27th July Morning Shift Let$$f: \mathbb{R} \rightarrow \mathbb{R}$$be a function defined as$$f(x)=a \sin \left(\frac{\pi[x]}{2}\right)+[2-x],...
JEE Main 2022 (Online) 27th July Morning Shift
The area bounded by the curves $$y=\left|x^{2}-1\right|$$ and $$y=1$$ is
JEE Main 2022 (Online) 26th July Evening Shift
$$\int\limits_{0}^{20 \pi}(|\sin x|+|\cos x|)^{2} d x \text { is equal to }$$
JEE Main 2022 (Online) 26th July Evening Shift
If $$a = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{{2n} \over {{n^2} + {k^2}}}}$$ and $$f(x) = \sq... JEE Main 2022 (Online) 26th July Morning Shift The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = ya is$${{364} \over 3}$$... JEE Main 2022 (Online) 26th July Morning Shift Let a smooth curve$$y=f(x)$$be such that the slope of the tangent at any point$$(x, y)$$on it is directly proportion... JEE Main 2022 (Online) 25th July Evening Shift Let$$[t]$$denote the greatest integer less than or equal to$$t$$. Then the value of the integral$$\int_{-3}^{101}\le...
JEE Main 2022 (Online) 25th July Evening Shift
Let the locus of the centre $$(\alpha, \beta), \beta>0$$, of the circle which touches the circle $$x^{2}+(y-1)^{2}=1$$ e...
JEE Main 2022 (Online) 25th July Morning Shift
The slope of the tangent to a curve $$C: y=y(x)$$ at any point $$(x, y)$$ on it is $$\frac{2 \mathrm{e}^{2 x}-6 \mathrm{... JEE Main 2022 (Online) 25th July Morning Shift For any real number$$x$$, let$$[x]$$denote the largest integer less than equal to$$x$$. Let$$f$$be a real valued f... JEE Main 2022 (Online) 25th July Morning Shift The area of the region given by$$A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}$$is : JEE Main 2022 (Online) 25th July Morning Shift Let$${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R$$, represents a circle with center ($$\alpha...
JEE Main 2022 (Online) 30th June Morning Shift
$$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}}$$ is equal to :...
JEE Main 2022 (Online) 30th June Morning Shift
The value of the integral $$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}} ... JEE Main 2022 (Online) 24th June Evening Shift The value of$$\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx} ...
JEE Main 2022 (Online) 25th June Morning Shift
The area of the region enclosed between the parabolas y2 = 2x $$-$$ 1 and y2 = 4x $$-$$ 3 is...
JEE Main 2022 (Online) 25th June Evening Shift
The area of the region bounded by y2 = 8x and y2 = 16(3 $$-$$ x) is equal to:
JEE Main 2022 (Online) 26th June Evening Shift
The area bounded by the curve y = |x2 $$-$$ 9| and the line y = 3 is :
JEE Main 2022 (Online) 26th June Morning Shift
The value of the integral $$\int\limits_{ - 2}^2 {{{|{x^3} + x|} \over {({e^{x|x|}} + 1)}}dx}$$ is equal to :...
JEE Main 2022 (Online) 27th June Morning Shift
The integral $$\int\limits_0^1 {{1 \over {{7^{\left[ {{1 \over x}} \right]}}}}dx}$$, where [ . ] denotes the greatest i...
JEE Main 2022 (Online) 27th June Evening Shift
Let f be a differentiable function in $$\left( {0,{\pi \over 2}} \right)$$. If $$\int\limits_{\cos x}^1 {{t^2}\,f(t)dt ... JEE Main 2022 (Online) 27th June Evening Shift If m and n respectively are the number of local maximum and local minimum points of the function$$f(x) = \int\limits_0^...
JEE Main 2022 (Online) 27th June Evening Shift
The area of the region S = {(x, y) : y2 $$\le$$ 8x, y $$\ge$$ $$\sqrt2$$x, x $$\ge$$ 1} is
JEE Main 2022 (Online) 28th June Morning Shift
Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral $$\int\limits_0^1 {[ - 8{x^... JEE Main 2022 (Online) 28th June Morning Shift Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2$$\tan x(\cos x - y)$$. If the curve passes thr... JEE Main 2022 (Online) 28th June Evening Shift The area of the bounded region enclosed by the curve$$y = 3 - \left| {x - {1 \over 2}} \right| - |x + 1|$$and the x-ax... JEE Main 2022 (Online) 28th June Evening Shift Let f : R$$\to$$R be a continuous function satisfying f(x) + f(x + k) = n, for all x$$\in$$R where k > 0 and n is a... JEE Main 2022 (Online) 28th June Evening Shift Let f : R$$\to$$R be a differentiable function such that$$f\left( {{\pi \over 4}} \right) = \sqrt 2 ,\,f\left( {{\pi...
JEE Main 2022 (Online) 28th June Evening Shift
If $$\int\limits_0^2 {\left( {\sqrt {2x} - \sqrt {2x - {x^2}} } \right)dx = \int\limits_0^1 {\left( {1 - \sqrt {1 - {y^... JEE Main 2022 (Online) 29th June Evening Shift Let f be a real valued continuous function on [0, 1] and$$f(x) = x + \int\limits_0^1 {(x - t)f(t)dt} $$. Then, which of... JEE Main 2022 (Online) 29th June Evening Shift$$\int_0^5 {\cos \left( {\pi \left( {x - \left[ {{x \over 2}} \right]} \right)} \right)dx} $$, where [t] denotes greates... JEE Main 2022 (Online) 29th June Morning Shift The area enclosed by y2 = 8x and y =$$\sqrt2$$x that lies outside the triangle formed by y =$$\sqrt2$$x, x = 1, y = ... JEE Main 2022 (Online) 29th June Morning Shift Let$$f:R \to R$$be a function defined by :$$f(x) = \left\{ {\matrix{ {\max \,\{ {t^3} - 3t\} \,t \le x} & ; & {x \...
JEE Main 2022 (Online) 29th June Morning Shift
The function f(x), that satisfies the condition $$f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy}$$, is :...
JEE Main 2021 (Online) 1st September Evening Shift
The area, enclosed by the curves $$y = \sin x + \cos x$$ and $$y = \left| {\cos x - \sin x} \right|$$ and the lines $$x ... JEE Main 2021 (Online) 1st September Evening Shift Let f : R$$\to$$R be a continuous function. Then$$\mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\...
JEE Main 2021 (Online) 1st September Evening Shift
If [x] is the greatest integer $$\le$$ x, then $${\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x... JEE Main 2021 (Online) 31st August Evening Shift Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If$$\int_0^x {\sqrt {1 - {{(f'(t))}^2}} ...
JEE Main 2021 (Online) 31st August Morning Shift
The value of the integral $$\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}}$$ is :
JEE Main 2021 (Online) 27th August Evening Shift
The area of the region bounded by the parabola (y $$-$$ 2)2 = (x $$-$$ 1), the tangent to it at the point whose ordinate...
JEE Main 2021 (Online) 27th August Evening Shift
$$\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx}$$ is equal to :
JEE Main 2021 (Online) 27th August Morning Shift
If $${U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2... JEE Main 2021 (Online) 27th August Morning Shift The value of$$\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}...
JEE Main 2021 (Online) 26th August Evening Shift
If the value of the integral $$\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta }$$, wh...
JEE Main 2021 (Online) 26th August Evening Shift
The value of $$\int\limits_{{{ - 1} \over {\sqrt 2 }}}^{{1 \over {\sqrt 2 }}} {{{\left( {{{\left( {{{x + 1} \over {x - 1... JEE Main 2021 (Online) 26th August Morning Shift If the area of the bounded region$$R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2...
JEE Main 2021 (Online) 27th July Morning Shift
The value of the definite integral$$\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({... JEE Main 2021 (Online) 27th July Morning Shift The value of$$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{j = 1}^n {{{(2j - 1) + 8n} \over {(2j - 1)...
JEE Main 2021 (Online) 27th July Morning Shift
The value of the integral $$\int\limits_{ - 1}^1 {\log \left( {x + \sqrt {{x^2} + 1} } \right)dx}$$ is :
JEE Main 2021 (Online) 25th July Evening Shift
If $$f(x) = \left\{ {\matrix{ {\int\limits_0^x {\left( {5 + \left| {1 - t} \right|} \right)dt,} } &amp; {x &gt; 2} \... JEE Main 2021 (Online) 25th July Evening Shift Let f : (a, b)$$\to$$R be twice differentiable function such that$$f(x) = \int_a^x {g(t)dt} $$for a differentiable f... JEE Main 2021 (Online) 27th July Evening Shift The area of the region bounded by y$$-$$x = 2 and x2 = y is equal to : JEE Main 2021 (Online) 27th July Evening Shift Let$$f:[0,\infty ) \to [0,\infty )$$be defined as$$f(x) = \int_0^x {[y]dy} $$where [x] is the greatest integer less t... JEE Main 2021 (Online) 25th July Morning Shift The area (in sq. units) of the region, given by the set$$\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\} $$i... JEE Main 2021 (Online) 25th July Morning Shift The value of the definite integral$$\int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}} $$is ... JEE Main 2021 (Online) 25th July Morning Shift If$$\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \righ...
JEE Main 2021 (Online) 22th July Evening Shift
Let $$g(t) = \int_{ - \pi /2}^{\pi /2} {\cos \left( {{\pi \over 4}t + f(x)} \right)} dx$$, where $$f(x) = {\log _e}\lef... JEE Main 2021 (Online) 20th July Evening Shift Let y = y(x) satisfies the equation$${{dy} \over {dx}} - |A| = 0$$, for all x &gt; 0, where$$A = \left[ {\matrix{ y...
JEE Main 2021 (Online) 20th July Evening Shift
If the real part of the complex number $${(1 - \cos \theta + 2i\sin \theta )^{ - 1}}$$ is $${1 \over 5}$$ for $$\theta ... JEE Main 2021 (Online) 20th July Evening Shift If [x] denotes the greatest integer less than or equal to x, then the value of the integral$$\int_{ - \pi /2}^{\pi /2} ...
JEE Main 2021 (Online) 20th July Evening Shift
The value of the integral $$\int\limits_{ - 1}^1 {{{\log }_e}(\sqrt {1 - x} + \sqrt {1 + x} )dx}$$ is equal to:
JEE Main 2021 (Online) 20th July Morning Shift
Let a be a positive real number such that $$\int_0^a {{e^{x - [x]}}} dx = 10e - 9$$ where [ x ] is the greatest integer ...
JEE Main 2021 (Online) 20th July Morning Shift
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) $... JEE Main 2021 (Online) 18th March Evening Shift The area bounded by the curve 4y2 = x2(4 $$-$$ x)(x $$-$$ 2) is equal to : JEE Main 2021 (Online) 18th March Evening Shift If the integral $$\int_0^{10} {{{[\sin 2\pi x]} \over {{e^{x - [x]}}}}} dx = \alpha {e^{ - 1}} + \beta {e^{ - {1 \over ... JEE Main 2021 (Online) 17th March Evening Shift Let f : R$$ \to $$R be defined as f(x) = e$$-$$xsinx. If F : [0, 1]$$ \to $$R is a differentiable function with that... JEE Main 2021 (Online) 17th March Evening Shift Which of the following statements is incorrect for the function g($$\alpha$$) for$$\alpha\in$$R such that$$g(\al... JEE Main 2021 (Online) 17th March Morning Shift 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) le... JEE Main 2021 (Online) 16th March Evening Shift Consider the integral $$I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx}$$, where [x] denotes the greatest inte... JEE Main 2021 (Online) 16th March Evening Shift For x &gt; 0, if $$f(x) = \int\limits_1^x {{{{{\log }_e}t} \over {(1 + t)}}dt}$$, then $$f(e) + f\left( {{1 \over e}} \... JEE Main 2021 (Online) 26th February Evening Shift The value of$$\sum\limits_{n = 1}^{100} {\int\limits_{n = 1}^n {{e^{x - [x]}}dx} } $$, where [ x ] is the greatest inte... JEE Main 2021 (Online) 26th February Morning Shift The value of$$\int\limits_{ - \pi /2}^{\pi /2} {{{{{\cos }^2}x} \over {1 + {3^x}}}} dx$$is : JEE Main 2021 (Online) 26th February Morning Shift If$${I_n} = \int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\cot }^n}x\,dx} $$, then : JEE Main 2021 (Online) 25th February Evening Shift The value of$$\int\limits_{ - 1}^1 {{x^2}{e^{[{x^3}]}}} dx$$, where [ t ] denotes the greatest integer$$ \le $$t, is ... JEE Main 2021 (Online) 25th February Morning Slot The area of the region :$$R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\} $$is : JEE Main 2021 (Online) 24th February Evening Slot Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2$$-$$x) for all x$$ \in $$(0, 2), f(0) ... JEE Main 2021 (Online) 24th February Evening Slot The value of the integral,$$\int\limits_1^3 {[{x^2} - 2x - 2]dx} $$, where [x] denotes the greatest integer less than o... JEE Main 2021 (Online) 24th February Evening Slot If a curve y = f(x) passes through the point (1, 2) and satisfies$$x {{dy} \over {dx}} + y = b{x^4}$$, then for what va... JEE Main 2021 (Online) 24th February Evening Slot The area (in sq. units) of the part of the circle x2 + y2 = 36, which is outside the parabola y2 = 9x, is :... JEE Main 2021 (Online) 24th February Morning Slot$$\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {\left( {\sin \sqrt t } \right)dt} } \over {{x^3}}}$$is equa... JEE Main 2021 (Online) 24th February Morning Slot The area (in sq. units) of the region enclosed by the curves y = x2 – 1 and y = 1 – x2 is equal to :... JEE Main 2020 (Online) 6th September Evening Slot The integral$$\int\limits_1^2 {{e^x}.{x^x}\left( {2 + {{\log }_e}x} \right)} dx$$equals : JEE Main 2020 (Online) 6th September Evening Slot If I1 =$$\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx$$and I2 =$$\int\limits_0^1 {{{\left( {1 - {x^{5... JEE Main 2020 (Online) 6th September Morning Slot The area (in sq. units) of the region A = {(x, y) : |x| + |y| $$\le$$ 1, 2y2 $$\ge$$ |x|} JEE Main 2020 (Online) 6th September Morning Slot The area (in sq. units) of the region A = {(x, y) : (x – 1)[x] $$\le$$ y $$\le$$ 2$$\sqrt x$$, 0 $$\le$$ x $$\l... JEE Main 2020 (Online) 5th September Evening Slot The value of$$\int\limits_{{{ - \pi } \over 2}}^{{\pi \over 2}} {{1 \over {1 + {e^{\sin x}}}}dx} $$is: JEE Main 2020 (Online) 5th September Morning Slot The integral$$\int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{\tan }^3}x.{{\sin }^2}3x\left( {2{{\sec }^2}x.{{\sin }^2... JEE Main 2020 (Online) 4th September Evening Slot Let $$f(x) = \left| {x - 2} \right|$$ and g(x) = f(f(x)), $$x \in \left[ {0,4} \right]$$. Then $$\int\limits_0^3 {\left(... JEE Main 2020 (Online) 4th September Morning Slot Let$$f\left( x \right) = \int {{{\sqrt x } \over {{{\left( {1 + x} \right)}^2}}}dx\left( {x \ge 0} \right)} $$. Then f... JEE Main 2020 (Online) 4th September Morning Slot If x3dy + xy dx = x2dy + 2y dx; y(2) = e and x &gt; 1, then y(4) is equal to : JEE Main 2020 (Online) 3rd September Evening Slot Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If T = {x$$ \in $$R | f(x) = f(0)}, ... JEE Main 2020 (Online) 3rd September Evening Slot If the value of the integral$$\int\limits_0^{{1 \over 2}} {{{{x^2}} \over {{{\left( {1 - {x^2}} \right)}^{{3 \over 2}}... JEE Main 2020 (Online) 3rd September Evening Slot $$\int\limits_{ - \pi }^\pi {\left| {\pi - \left| x \right|} \right|dx}$$ is equal to : JEE Main 2020 (Online) 3rd September Morning Slot The area (in sq. units) of the region { (x, y) : 0 $$\le$$ y $$\le$$ x2 + 1, 0 $$\le$$ y $$\le$$ x + 1, $${1 \o... JEE Main 2020 (Online) 3rd September Morning Slot Consider a region R = {(x, y)$$ \in $$R : x2$$ \le $$y$$ \le $$2x}. if a line y =$$\alpha $$divides the area of ... JEE Main 2020 (Online) 2nd September Evening Slot Let y = y(x) be the solution of the differential equation,$${{2 + \sin x} \over {y + 1}}.{{dy} \over {dx}} = - \cos x$...
JEE Main 2020 (Online) 2nd September Morning Slot
Area (in sq. units) of the region outside $${{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1$$ and inside...
JEE Main 2020 (Online) 2nd September Morning Slot
Given : $$f(x) = \left\{ {\matrix{ {x\,\,\,\,\,,} &amp; {0 \le x &lt; {1 \over 2}} \cr {{1 \over 2}\,\,\,\,,} &a... JEE Main 2020 (Online) 9th January Evening Slot If for all real triplets (a, b, c), ƒ(x) = a + bx + cx2; then$$\int\limits_0^1 {f(x)dx} $$is equal to : JEE Main 2020 (Online) 9th January Morning Slot The value of$$\int\limits_0^{2\pi } {{{x{{\sin }^8}x} \over {{{\sin }^8}x + {{\cos }^8}x}}} dx$$is equal to : JEE Main 2020 (Online) 9th January Morning Slot If$$I = \int\limits_1^2 {{{dx} \over {\sqrt {2{x^3} - 9{x^2} + 12x + 4} }}} $$, then : JEE Main 2020 (Online) 8th January Evening Slot The area (in sq. units) of the region {(x,y)$$ \in $$R2 : x2$$ \le $$y$$ \le $$3 – 2x}, is... JEE Main 2020 (Online) 8th January Evening Slot For a &gt; 0, let the curves C1 : y2 = ax and C2 : x2 = ay intersect at origin O and a point P. Let the line x = b (0 &l... JEE Main 2020 (Online) 8th January Morning Slot Let ƒ(x) = (sin(tan–1x) + sin(cot–1x))2 – 1, |x| &gt; 1. If$${{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\si...
JEE Main 2020 (Online) 8th January Morning Slot
If $$\theta$$1 and $$\theta$$2 be respectively the smallest and the largest values of $$\theta$$ in (0, 2$$\pi$$) ...
JEE Main 2020 (Online) 7th January Evening Slot
The area (in sq. units) of the region {(x, y) $$\in$$ R2 | 4x2 $$\le$$ y $$\le$$ 8x + 12} is :...
JEE Main 2020 (Online) 7th January Evening Slot
The value of $$\alpha$$ for which $$4\alpha \int\limits_{ - 1}^2 {{e^{ - \alpha \left| x \right|}}dx} = 5$$, is:
JEE Main 2020 (Online) 7th January Evening Slot
If ƒ(a + b + 1 - x) = ƒ(x), for all x, where a and b are fixed positive real numbers, then $${1 \over {a + b}}\int_a^b {... JEE Main 2020 (Online) 7th January Morning Slot The area of the region, enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 ... JEE Main 2020 (Online) 7th January Morning Slot A value of$$\alpha $$such that$$\int\limits_\alpha ^{\alpha + 1} {{{dx} \over {\left( {x + \alpha } \right)\left( {x...
JEE Main 2019 (Online) 12th April Evening Slot
If the area (in sq. units) bounded by the parabola y2 = 4$$\lambda$$x and the line y = $$\lambda$$x, $$\lambda$$ &gt...
JEE Main 2019 (Online) 12th April Evening Slot
Let f : R $$\to$$ R be a continuously differentiable function such that f(2) = 6 and f'(2) = $${1 \over {48}}$$. If $$... JEE Main 2019 (Online) 12th April Morning Slot If the area (in sq. units) of the region {(x, y) : y2$$ \le $$4x, x + y$$ \le $$1, x$$ \ge $$0, y$$ \ge $$0} is ... JEE Main 2019 (Online) 12th April Morning Slot If$$\int\limits_0^{{\pi \over 2}} {{{\cot x} \over {\cot x + \cos ecx}}} dx$$= m($$\pi $$+ n), then m.n is equal to JEE Main 2019 (Online) 12th April Morning Slot The integral$$\int\limits_{\pi /6}^{\pi /3} {{{\sec }^{2/3}}} x\cos e{c^{4/3}}xdx$$is equal to : JEE Main 2019 (Online) 10th April Evening Slot The area (in sq.units) of the region bounded by the curves y = 2x and y = |x + 1|, in the first quadrant is : ... JEE Main 2019 (Online) 10th April Evening Slot The value of$$\int\limits_0^{2\pi } {\left[ {\sin 2x\left( {1 + \cos 3x} \right)} \right]} dx$$, where [t] denotes the ... JEE Main 2019 (Online) 10th April Morning Slot The area (in sq. units) of the region A = {(x, y) :$${{y{}^2} \over 2} \le $$x$$ \le $$y + 4} is :- JEE Main 2019 (Online) 9th April Evening Slot If f : R$$ \to $$R is a differentiable function and f(2) = 6, then$$\mathop {\lim }\limits_{x \to 2} {{\int\limits_6^...
JEE Main 2019 (Online) 9th April Evening Slot
The value of the integral $$\int\limits_0^1 {x{{\cot }^{ - 1}}(1 - {x^2} + {x^4})dx}$$ is :-
JEE Main 2019 (Online) 9th April Evening Slot
The area (in sq. units) of the region A = {(x, y) : x2 $$\le$$ y $$\le$$ x + 2} is
JEE Main 2019 (Online) 9th April Morning Slot
The value of $$\int\limits_0^{\pi /2} {{{{{\sin }^3}x} \over {\sin x + \cos x}}dx}$$ is
JEE Main 2019 (Online) 9th April Morning Slot
Let S($$\alpha$$) = {(x, y) : y2 $$\le$$ x, 0 $$\le$$ x $$\le$$ $$\alpha$$} and A($$\alpha$$) is area of the r...
JEE Main 2019 (Online) 8th April Evening Slot
The area (in sq. units) of the region A = { (x, y) $$\in$$ R × R|&nbsp;&nbsp;0 $$\le$$ x $$\le$$ 3, 0 $$\le$$ y ...
JEE Main 2019 (Online) 8th April Morning Slot
If $$f(x) = {{2 - x\cos x} \over {2 + x\cos x}}$$ and g(x) = logex, (x &gt; 0) then the value of integral $$\int\limits_... JEE Main 2019 (Online) 8th April Morning Slot The integral$$\int\limits_1^e {\left\{ {{{\left( {{x \over e}} \right)}^{2x}} - {{\left( {{e \over x}} \right)}^x}} \ri...
JEE Main 2019 (Online) 12th January Evening Slot
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...
JEE Main 2019 (Online) 12th January Morning Slot
The integral&nbsp;&nbsp;$$\int\limits_{\pi /6}^{\pi /4} {{{dx} \over {\sin 2x\left( {{{\tan }^5}x + {{\cot }^5}x} \right... JEE Main 2019 (Online) 11th January Evening Slot The area (in sq. units) in the first quadrant bounded by the parabola, y = x2 + 1, the tangent to it at the point (2, 5)... JEE Main 2019 (Online) 11th January Evening Slot The area (in sq. units) of the region bounded by the curve x2 = 4y and the straight line x = 4y – 2 is : JEE Main 2019 (Online) 11th January Morning Slot If&nbsp;&nbsp;$$\int {{{\sqrt {1 - {x^2}} } \over {{x^4}}}} $$dx = A(x)$${\left( {\sqrt {1 - {x^2}} } \right)^m}$$+ C,... JEE Main 2019 (Online) 11th January Morning Slot The value of the integral$$\int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}...
JEE Main 2019 (Online) 11th January Morning Slot
If&nbsp;&nbsp;$$\int\limits_0^x \,$$f(t) dt = x2 + $$\int\limits_x^1 \,$$ t2f(t) dt then f '$$\left( {{1 \over 2}} \... JEE Main 2019 (Online) 10th January Evening Slot The value of &nbsp;&nbsp;$$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {\left[ x \right] + \left[ {\sin x} \right] + 4...
JEE Main 2019 (Online) 10th January Evening Slot
If the area enclosed between the curves y = kx2 and x = ky2, (k &gt; 0), is 1 square unit. Then k is -...
JEE Main 2019 (Online) 10th January Morning Slot
Let &nbsp;$${\rm I} = \int\limits_a^b {\left( {{x^4} - 2{x^2}} \right)} dx.$$&nbsp;&nbsp;If I is minimum then the ordere...
JEE Main 2019 (Online) 10th January Morning Slot
If&nbsp;&nbsp;&nbsp;$$\int\limits_0^{{\pi \over 3}} {{{\tan \theta } \over {\sqrt {2k\,\sec \theta } }}} \,d\theta = 1... JEE Main 2019 (Online) 9th January Evening Slot The area of the region A = {(x, y) : 0$$ \le $$y$$ \le $$x |x| + 1&nbsp;&nbsp;and&nbsp;&nbsp;$$-$$1$$ \le $$x$$ \...
JEE Main 2019 (Online) 9th January Evening Slot
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 :...
JEE Main 2019 (Online) 9th January Morning Slot
The value of $$\int\limits_0^\pi {{{\left| {\cos x} \right|}^3}} \,dx$$ is :
JEE Main 2019 (Online) 9th January Morning Slot
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 &gt;1) is...
JEE Main 2018 (Online) 16th April Morning Slot
If $$f(x) = \int\limits_0^x {t\left( {\sin x - \sin t} \right)dt\,\,\,}$$ then :
JEE Main 2018 (Online) 16th April Morning Slot
If &nbsp;&nbsp;$${I_1} = \int_0^1 {{e^{ - x}}} {\cos ^2}x{\mkern 1mu} dx;$$ &nbsp;&nbsp; $${I_2} = \int_0^1 {{e^{ - {x^... JEE Main 2018 (Online) 15th April Evening Slot The value of integral$$\int_{{\pi \over 4}}^{{{3\pi } \over 4}} {{x \over {1 + \sin x}}dx} $$is : JEE Main 2018 (Online) 15th April Evening Slot The value of the integral$$\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {{{\sin }^4}} x\left( {1 + \log \left( {{{2...
JEE Main 2018 (Online) 15th April Morning Slot
The area (in sq. units) of the region {x $$\in$$ R : x $$\ge$$ 0, y $$\ge$$ 0, y $$\ge$$ x $$-$$ 2 &nbsp;and y...
JEE Main 2018 (Online) 15th April Morning Slot
The value of $$\int\limits_{ - \pi /2}^{\pi /2} {{{{{\sin }^2}x} \over {1 + {2^x}}}} dx$$ is
JEE Main 2018 (Offline)
Let g(x) = cosx2, f(x) = $$\sqrt x$$ and $$\alpha ,\beta \left( {\alpha &lt; \beta } \right)$$ be the roots of the qua...
JEE Main 2018 (Offline)
If &nbsp;&nbsp; $$\int\limits_1^2 {{{dx} \over {{{\left( {{x^2} - 2x + 4} \right)}^{{3 \over 2}}}}}} = {k \over {k + 5}... JEE Main 2017 (Online) 9th April Morning Slot If &nbsp;&nbsp;$$\mathop {\lim }\limits_{n \to \infty } \,\,{{{1^a} + {2^a} + ...... + {n^a}} \over {{{(n + 1)}^{a - 1}...
JEE Main 2017 (Online) 9th April Morning Slot
The area (in sq. units) of the smaller portion enclosed between the curves, x2 + y2 = 4 and y2 = 3x, is : ...
JEE Main 2017 (Online) 8th April Morning Slot
The integral $$\int_{{\pi \over {12}}}^{{\pi \over 4}} {\,\,{{8\cos 2x} \over {{{\left( {\tan x + \cot x} \right)}^3}}... JEE Main 2017 (Online) 8th April Morning Slot The area (in sq. units) of the region$$\left\{ {\left( {x,y} \right):x \ge 0,x + y \le 3,{x^2} \le 4y\,and\,y \le 1 + \...
JEE Main 2017 (Offline)
The integral $$\int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{dx} \over {1 + \cos x}}}$$ is equal to
JEE Main 2017 (Offline)
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 +...
JEE Main 2016 (Online) 10th April Morning Slot
The value of the integral $$\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + ... JEE Main 2016 (Online) 10th April Morning Slot The area (in sq. units) of the region described by A= {(x, y)$$\left| {} \right.$$y$$ \ge $$x2$$-$$5x + 4, x + y$$ ...
JEE Main 2016 (Online) 9th April Morning Slot
If&nbsp;&nbsp; $$2\int\limits_0^1 {{{\tan }^{ - 1}}xdx = \int\limits_0^1 {{{\cot }^{ - 1}}} } \left( {1 - x + {x^2}} \ri... JEE Main 2016 (Online) 9th April Morning Slot The area (in sq. units) of the region$$\left\{ {\left( {x,y} \right):{y^2} \ge 2x\,\,\,and\,\,\,{x^2} + {y^2} \le 4x,x ...
JEE Main 2016 (Offline)
The area (in sq. units) of the region described by $$\left\{ {\left( {x,y} \right):{y^2} \le 2x} \right.$$ and $$\left.... JEE Main 2015 (Offline) The integral$$\int\limits_2^4 {{{\log \,{x^2}} \over {\log {x^2} + \log \left( {36 - 12x + {x^2}} \right)}}dx} $$is... JEE Main 2015 (Offline) The integral$$\int\limits_0^\pi {\sqrt {1 + 4{{\sin }^2}{x \over 2} - 4\sin {x \over 2}{\mkern 1mu} } } dx$$equals: JEE Main 2014 (Offline) The area of the region described by$$A = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right.$$and$$\left. {{y...
JEE Main 2014 (Offline)
The area (in square units) bounded by the curves $$y = \sqrt {x,} 2y - x + 3 = 0,$$ $$x$$-axis, and lying in the first q...
JEE Main 2013 (Offline)
Statement-1 : The value of the integral $$\int\limits_{\pi /6}^{\pi /3} {{{dx} \over {1 + \sqrt {\tan \,x} }}}$$ is eq...
JEE Main 2013 (Offline)
The area between the parabolas $${x^2} = {y \over 4}$$ and $${x^2} = 9y$$ and the straight line $$y=2$$ is :
AIEEE 2012
The area of the region enclosed by the curves $$y = x,x = e,y = {1 \over x}$$ and the positive $$x$$-axis is
AIEEE 2011
The value of $$\int\limits_0^1 {{{8\log \left( {1 + x} \right)} \over {1 + {x^2}}}} dx$$ is
AIEEE 2011
The area bounded by the curves $$y = \cos x$$ and $$y = \sin x$$ between the ordinates $$x=0$$ and $$x = {{3\pi } \over ... AIEEE 2010 Let$$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 ...
AIEEE 2010
$$\int\limits_0^\pi {\left[ {\cot x} \right]dx,}$$ where $$\left[ . \right]$$ denotes the greatest integer function, i...
AIEEE 2009
The area of the region bounded by the parabola $${\left( {y - 2} \right)^2} = x - 1,$$ the tangent of the parabola at th...
AIEEE 2009
The area of the plane region bounded by the curves $$x + 2{y^2} = 0$$ and $$\,x + 3{y^2} = 1$$ is equal to
AIEEE 2008
The area enclosed between the curves $${y^2} = x$$ and $$y = \left| x \right|$$ is
AIEEE 2007
Let $$I = \int\limits_0^1 {{{\sin x} \over {\sqrt x }}dx}$$ and $$J = \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx} .... AIEEE 2007 The solution for$$x$$of the equation$$\int\limits_{\sqrt 2 }^x {{{dt} \over {t\sqrt {{t^2} - 1} }} = {\pi \over 2}} ...
AIEEE 2007
Let $$F\left( x \right) = f\left( x \right) + f\left( {{1 \over x}} \right),$$ where $$f\left( x \right) = \int\limits_l... AIEEE 2007 The value of$$\int\limits_1^a {\left[ x \right]} f'\left( x \right)dx,a &gt; 1$$where$${\left[ x \right]}$$denotes t... AIEEE 2006$$\int\limits_{ - {{3\pi } \over 2}}^{ - {\pi \over 2}} {\left[ {{{\left( {x + \pi } \right)}^3} + {{\cos }^2}\left( {x...
AIEEE 2006
$$\int\limits_0^\pi {xf\left( {\sin x} \right)dx}$$ is equal to
AIEEE 2006
The value of integral, $$\int\limits_3^6 {{{\sqrt x } \over {\sqrt {9 - x} + \sqrt x }}} dx$$ is
AIEEE 2005
The value of $$\int\limits_{ - \pi }^\pi {{{{{\cos }^2}} \over {1 + {a^x}}}dx,\,\,a &gt; 0,}$$ is
AIEEE 2005
Let $$f(x)$$ be a non - negative continuous function such that the area bounded by the curve $$y=f(x),$$ $$x$$-axis and...
AIEEE 2005
The parabolas $${y^2} = 4x$$ and $${x^2} = 4y$$ divide the square region bounded by the lines $$x=4,$$ $$y=4$$ and the c...
AIEEE 2005
The area enclosed between the curve $$y = {\log _e}\left( {x + e} \right)$$ and the coordinate axes is
AIEEE 2005
If $${I_1} = \int\limits_0^1 {{2^{{x^2}}}dx,{I_2} = \int\limits_0^1 {{2^{{x^3}}}dx,\,{I_3} = \int\limits_1^2 {{2^{{x^2}}... AIEEE 2005 The area of the region bounded by the curves$$y = \left| {x - 2} \right|,x = 1,x = 3$$and the$$x$$-axis is AIEEE 2004 If$$f\left( x \right) = {{{e^x}} \over {1 + {e^x}}},{I_1} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {x...
AIEEE 2004
If $$\int\limits_0^\pi {xf\left( {\sin x} \right)dx = A\int\limits_0^{\pi /2} {f\left( {\sin x} \right)dx,} }$$ then...
AIEEE 2004
The value of $$I = \int\limits_0^{\pi /2} {{{{{\left( {\sin x + \cos x} \right)}^2}} \over {\sqrt {1 + \sin 2x} }}dx}$$...
AIEEE 2004
The value of $$\int\limits_{ - 2}^3 {\left| {1 - {x^2}} \right|dx}$$ is
AIEEE 2004
$$\mathop {Lim}\limits_{n \to \infty } \sum\limits_{r = 1}^n {{1 \over n}{e^{{r \over n}}}}$$ is
AIEEE 2004
The value of the integral $$I = \int\limits_0^1 {x{{\left( {1 - x} \right)}^n}dx}$$ is
AIEEE 2003
If $$f\left( {a + b - x} \right) = f\left( x \right)$$ then $$\int\limits_a^b {xf\left( x \right)dx}$$ is equal to
AIEEE 2003
Let $$f(x)$$ be a function satisfying $$f'(x)=f(x)$$ with $$f(0)=1$$ and $$g(x)$$ be a function that satisfies $$f\left(... AIEEE 2003 The area of the region bounded by the curves$$y = \left| {x - 1} \right|$$and$$y = 3 - \left| x \right|$$is AIEEE 2003 The area bounded by the curves$$y = \ln x,y = \ln \left| x \right|,y = \left| {\ln {\mkern 1mu} x} \right|$$and$$y = ...
AIEEE 2002
If $$y=f(x)$$ makes +$$ve$$ intercept of $$2$$ and $$0$$ unit on $$x$$ and $$y$$ axes and encloses an area of $$3/4$$ sq...
AIEEE 2002
$$\int_{ - \pi }^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} dx$$ is
AIEEE 2002
$$\int\limits_0^2 {\left[ {{x^2}} \right]dx}$$ is
AIEEE 2002
$${I_n} = \int\limits_0^{\pi /4} {{{\tan }^n}x\,dx}$$ then $$\,\mathop {\lim }\limits_{n \to \infty } \,n\left[ {{I_n} ... AIEEE 2002$$\int_0^{10\pi } {\left| {\sin x} \right|dx} $$is AIEEE 2002 ## Numerical More The value of the integral$$\int\limits_{0}^{\frac{\pi}{2}} 60 \frac{\sin (6 x)}{\sin x} d x$$is equal to _________.... JEE Main 2022 (Online) 28th July Evening Shift If$$\int\limits_{0}^{\sqrt{3}} \frac{15 x^{3}}{\sqrt{1+x^{2}+\sqrt{\left(1+x^{2}\right)^{3}}}} \mathrm{~d} x=\alpha \sq...
JEE Main 2022 (Online) 28th July Morning Shift
Let f be a differentiable function satisfying $$f(x)=\frac{2}{\sqrt{3}} \int\limits_{0}^{\sqrt{3}} f\left(\frac{\lambda^... JEE Main 2022 (Online) 27th July Evening Shift Let$$f(x)=\min \{[x-1],[x-2], \ldots,[x-10]\}$$where [t] denotes the greatest integer$$\leq \mathrm{t}$$. Then$$\int...
JEE Main 2022 (Online) 27th July Evening Shift
Let a curve $$y=y(x)$$ pass through the point $$(3,3)$$ and the area of the region under this curve, above the $$x$$-axi...
JEE Main 2022 (Online) 26th July Morning Shift
If $$\mathrm{n}(2 \mathrm{n}+1) \int_{0}^{1}\left(1-x^{\mathrm{n}}\right)^{2 \mathrm{n}} \mathrm{d} x=1177 \int_{0}^{1}\... JEE Main 2022 (Online) 26th July Morning Shift Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve$$4{x^3} - 3x{y^2} + 6{x^2} - 5xy - 8...
JEE Main 2022 (Online) 25th July Evening Shift
Let $${a_n} = \int_{ - 1}^n {\left( {1 + {x \over 2} + {{{x^2}} \over 3} + \,\,.....\,\, + \,\,{{{x^{n - 1}}} \over n}} ... JEE Main 2022 (Online) 25th July Evening Shift Let$$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) ...
JEE Main 2022 (Online) 30th June Morning Shift
If for some $$\alpha$$ > 0, the area of the region $$\{ (x,y):|x + \alpha | \le y \le 2 - |x|\}$$ is equal to $${3 \ove... JEE Main 2022 (Online) 30th June Morning Shift Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1, ... JEE Main 2022 (Online) 24th June Morning Shift Let$$\mathop {Max}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \alpha $$and$$\mathop ...
JEE Main 2022 (Online) 24th June Morning Shift
Let $$f(\theta ) = \sin \theta + \int\limits_{ - \pi /2}^{\pi /2} {(\sin \theta + t\cos \theta )f(t)dt}$$. Then the v...
JEE Main 2022 (Online) 24th June Morning Shift
The area (in sq. units) of the region enclosed between the parabola y2 = 2x and the line x + y = 4 is __________....
JEE Main 2022 (Online) 24th June Evening Shift
The value of b > 3 for which $$12\int\limits_3^b {{1 \over {({x^2} - 1)({x^2} - 4)}}dx = {{\log }_e}\left( {{{49} \over ... JEE Main 2022 (Online) 25th June Evening Shift The integral$${{24} \over \pi }\int_0^{\sqrt 2 } {{{(2 - {x^2})dx} \over {(2 + {x^2})\sqrt {4 + {x^4}} }}} $$is equal ... JEE Main 2022 (Online) 26th June Evening Shift The value of the integral$${{48} \over {{\pi ^4}}}\int\limits_0^\pi {\left( {{{3\pi {x^2}} \over 2} - {x^3}} \right){{...
JEE Main 2022 (Online) 26th June Morning Shift
Let f(x) = max {|x + 1|, |x + 2|, ....., |x + 5|}. Then $$\int\limits_{ - 6}^0 {f(x)dx}$$ is equal to __________....
JEE Main 2022 (Online) 26th June Morning Shift
Let $${A_1} = \left\{ {(x,y):|x| \le {y^2},|x| + 2y \le 8} \right\}$$ and $${A_2} = \left\{ {(x,y):|x| + |y| \le k} \rig... JEE Main 2022 (Online) 27th June Morning Shift If the area of the region$$\left\{ {(x,y):{x^{{2 \over 3}}} + {y^{{2 \over 3}}} \le 1,\,x + y \ge 0,\,y \ge 0} \right\}...
JEE Main 2022 (Online) 27th June Evening Shift
For real numbers a, b (a > b > 0), let Area $$\left\{ {(x,y):{x^2} + {y^2} \le {a^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{... JEE Main 2022 (Online) 29th June Evening Shift If the line y = mx bisects the area enclosed by the lines x = 0, y = 0, x =$${3 \over 2}$$and the curve y = 1 + 4x$$-...
JEE Main 2021 (Online) 31st August Evening Shift
If $$x\phi (x) = \int\limits_5^x {(3{t^2} - 2\phi '(t))dt}$$, x &gt; $$-$$2, and $$\phi$$(0) = 4, then $$\phi$$(2) is _...
JEE Main 2021 (Online) 31st August Morning Shift
Let [t] denote the greatest integer $$\le$$ t. Then the value of $$8.\int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx}$$ ...
JEE Main 2021 (Online) 31st August Morning Shift
Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x3 $$-$$ 3x2 $$-$$ 12x...
JEE Main 2021 (Online) 26th August Evening Shift
The area of the region $$S = \{ (x,y):3{x^2} \le 4y \le 6x + 24\}$$ is ____________.
JEE Main 2021 (Online) 26th August Morning Shift
Let the domain of the function$$f(x) = {\log _4}\left( {{{\log }_5}\left( {{{\log }_3}(18x - {x^2} - 77)} \right)} \righ... JEE Main 2021 (Online) 27th July Morning Shift Let a curve y = f(x) pass through the point (2, (loge2)2) and have slope$${{2y} \over {x{{\log }_e}x}}$$for all positi... JEE Main 2021 (Online) 25th July Evening Shift If$$\int_0^\pi {({{\sin }^3}x){e^{ - {{\sin }^2}x}}dx = \alpha - {\beta \over e}\int_0^1 {\sqrt t {e^t}dt} } $$, the... JEE Main 2021 (Online) 27th July Evening Shift The area (in sq. units) of the region bounded by the curves x2 + 2y$$-$$1 = 0, y2 + 4x$$-$$4 = 0 and y2$$-$$4x$$-...
JEE Main 2021 (Online) 22th July Evening Shift
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 tang...
JEE Main 2021 (Online) 20th July Morning Shift
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 maxim...
JEE Main 2021 (Online) 18th March Evening Shift
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 ...
JEE Main 2021 (Online) 18th March Morning Shift
Let f : [$$-$$3, 1] $$\to$$ R be given as $$f(x) = \left\{ \matrix{ \min \,\{ (x + 6),{x^2}\}, - 3 \le x \le 0 \hfil... JEE Main 2021 (Online) 17th March Evening Shift Let$${I_n} = \int_1^e {{x^{19}}{{(\log |x|)}^n}} dx$$, where n$$\in$$N. If (20)I10 =$$\alpha$$I9 +$$\beta$$I8, for na... JEE Main 2021 (Online) 17th March Evening Shift If [ . ] represents the greatest integer function, then the value of$$\left| {\int\limits_0^{\sqrt {{\pi \over 2}} } {...
JEE Main 2021 (Online) 17th March Morning Shift
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...
JEE Main 2021 (Online) 16th March Morning Shift
Let f : R $$\to$$ R be a continuous function such that f(x) + f(x + 1) = 2, for all x$$\in$$R. If $${I_1} = \int\limit... JEE Main 2021 (Online) 16th March Morning Shift Let the curve y = y(x) be the solution of the differential equation,$${{dy} \over {dx}}$$= 2(x + 1). If the numerical ... JEE Main 2021 (Online) 16th March Morning Shift If$${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^...
JEE Main 2021 (Online) 26th February Evening Shift
Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, $$... JEE Main 2021 (Online) 26th February Evening Shift The area bounded by the lines y = || x$$-$$1 |$$-$$2 | is ___________. JEE Main 2021 (Online) 26th February Morning Shift The value of the integral$$\int\limits_0^\pi {|{{\sin }^2}2x|dx} $$is ___________. JEE Main 2021 (Online) 26th February Morning Shift The value of$$\int\limits_{ - 2}^2 {|3{x^2} - 3x - 6|dx} $$is ___________. JEE Main 2021 (Online) 25th February Evening Shift The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points o... JEE Main 2021 (Online) 25th February Morning Slot If$$\int\limits_{ - a}^a {\left( {\left| x \right| + \left| {x - 2} \right|} \right)} dx = 22$$, (a &gt; 2) and [x] den... JEE Main 2021 (Online) 24th February Morning Slot Let {x} and [x] denote the fractional part of x and the greatest integer$$ \le $$x respectively of a real number x. If... JEE Main 2020 (Online) 4th September Evening Slot Let [t] denote the greatest integer less than or equal to t. Then the value of$$\int\limits_1^2 {\left| {2x - \left[ {3...
JEE Main 2020 (Online) 2nd September Evening Slot
The integral $$\int\limits_0^2 {\left| {\left| {x - 1} \right| - x} \right|dx}$$ is equal to______.
JEE Main 2020 (Online) 2nd September Morning Slot
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 ...
JEE Main 2020 (Online) 8th January Evening Slot

## MCQ (More than One Correct Answer)

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If $$g\left( x \right) = \int\limits_0^x {\cos 4t\,dt,}$$ then $$g\left( {x + \pi } \right)$$ equals
AIEEE 2012

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