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The area enclosed by the curves \( y=\log_e(x+e^2) \), \( x=\log_e\!\left(\dfrac{2}{y}\right) \) and \( x=\log_e 2 \), above the line \( y=1 \) is:

\(\displaystyle \lim_{n \to \infty} \frac{1}{2^n} \left( \frac{1}{\sqrt{1-\dfrac{1}{2^n}}} + \frac{1}{\sqrt{1-\dfrac{2}{2^n}}} + \frac{1}{\sqrt{1-\dfrac{3}{2^n}}} + \ldots + \frac{1}{\sqrt{1-\dfrac{2^n-1}{2^n}}} \right) \) is equal to:

Let \( I_n(x)=\displaystyle\int\limits_{0}^{x}\dfrac{1}{(t^2+5)^n}\,dt,\ n=1,2,3,\ldots \). Then:

The area bounded by the curves \( y = \left|x^{2}-1\right| \) and \( y = 1 \) is:

Let \( [t] \) denote the greatest integer less than or equal to \( t \). Then, the value of the integral \( \int_{0}^{1} \left[ -8x^2 + 6x - 1 \right] dx \) is equal to:

Let \( f(x) = \max\{\, |x+1|, |x+2|, \ldots, |x+5| \,\} \). Then \( \int_{-6}^{0} f(x)\,dx \) is equal to______________.

If \( x = x(y) \) is the solution of the differential equation \( y\,\frac{dx}{dy} = 2x + y^3 (y+1) e^{y} \), \( x(1) = 0 \); then \( x(e) \) is equal to:

If the area of the region \( \left\{(x,y): x^{\frac{2}{3}}+y^{\frac{2}{3}} \le 1, \; x+y \ge 0, \; y \ge 0 \right\} \) is \( A \), then \( \frac{256A}{\pi} \) is equal to __________.

Let \( [ t ]\) denote the greatest integer less than or equal to \( t \). Then the value of the integral \( \displaystyle\int\limits_{-3}^{101} \left( \big[ \sin(\pi x) \big] + e^{[\cos(2\pi x) ]} \right) dx \) is equal to:

The area of the region given by \( A=\{(x,y): x^2 \leq y \leq \min\{x+2,\, 4-3x\}\} \) is :

The integral \(\displaystyle \int\limits_{0}^{\dfrac{\pi}{2}} \frac{1}{3 + 2\sin x + \cos x} \, dx \) is equal to:

Let \( a_n =\displaystyle\int\limits_{-1}^{n} \left( 1 + \frac{x}{2} + \frac{x^2}{3} + \ldots + \frac{x^{n-1}}{n} \right) dx \) for every \( n \in \mathbb{N} \). Then the sum of all the elements of the set \( \{ n \in \mathbb{N} : a_n \in (2, 30) \} \) is ____________ .

A point \( P \) moves so that the sum of squares of its distances from the points \( (1, 2) \) and \( (-2, 1) \) is \( 14 \). Let \( f(x,y)=0 \) be the locus of \( P \), which intersects the x-axis at the points \( A, B \) and the y-axis at the points \( C, D \). Then the area of the quadrilateral \( ACBD \) is equal to:

The area of the smaller region enclosed by the curves \( y^{2}=8x+4 \) and \( x^{2}+y^{2}+4\sqrt{3}x-4=0 \) is equal to:

\( a=\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\dfrac{2n}{n^{2}+k^{2}} \) and \( f(x)=\sqrt{\dfrac{1-\cos x}{1+\cos x}},\, x\in(0,1), \) then:

Let \( f \) be a real valued continuous function on \( [0,1] \) and \( f(x)=x+\int_{0}^{1}(x-t)f(t)\,dt \). Then, which of the following points \( (x,y) \) lies on the curve \( y=f(x) \)?

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 \( I_{1} = \int_{0}^{4nk} f(x) \, dx \) and \( I_{2} = \int_{-k}^{3k} f(x) \, dx \), then:

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined as \( f(x) = a\sin\!\left(\dfrac{\pi [x]}{2}\right) + [2-x],\, a \in \mathbb{R} \), where \( [t] \) is the greatest integer less than or equal to \( t \). If \( \displaystyle\lim_{x\to -1} f(x) \) exists, then the value of \( \displaystyle\int\limits_{0}^{4} f(x)\,dx \) is equal to:

\( \displaystyle \lim_{n \to \infty} \left( \frac{n^{2}}{(n^{2}+1)(n+1)} + \frac{n^{2}}{(n^{2}+4)(n+2)} + \frac{n^{2}}{(n^{2}+9)(n+3)} + \ldots + \frac{n^{2}}{(n^{2}+n^{2})(n+n)} \right) \) is equal to:

The area enclosed by \( y^{2}=8x \) and \( y=\sqrt{2x} \) that lies outside the triangle formed by \( y=\sqrt{2x}, \ x=1, \ y=2\sqrt{2} \) is equal to:

The value of the integral \( \displaystyle \int_{-2}^{2}\dfrac{|x^{3}+x|}{e^{x|x|}+1}\,dx \) is equal to:

The value of \( \displaystyle \int_{0}^{\pi} \frac{e^{\cos x}\sin x}{\left(1+\cos^2 x\right)\left(e^{\cos x}+e^{-\cos x}\right)}\,dx \) is equal to:

The area bounded by the curve \( y=|x^{2}-9| \) and the line \( y=3 \) is:

The area of the region bounded by \( y^2 = 8x \) and \( y^2 = 16(3 - x) \) is equal to:

$\int_{0}^{5} \cos \left( \pi \left( x - \left[ \frac{x}{2} \right] \right) \right) \, dx$ , Where $[t]$ denotes greatest integer less than or equal to $t$, is equal to:

The value of \( b > 3 \) for which \( 12 \int_{3}^{b} \frac{1}{(x^2 - 1)(x^2 - 4)} \, dx = \log_e \!\left( \frac{49}{40} \right) \) is equal to ________.

If \( b_n = \int_{0}^{\pi/2} \frac{\cos^2(nx)}{\sin x} \, dx , \, n \in \mathbb{N} , \) then

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 interval \( [-10,10] \) by . Then the value of \( \dfrac{\pi^{2}}{10}\displaystyle\int\limits_{-10}^{10} f(x)\cos \pi x \, dx \) is :

Let \( f \) be a differentiable function in \( \left(0, \frac{\pi}{2}\right) \). If \( \int_{\cos x}^{1} t^{2} f(t)\, dt = \sin^{3} x + \cos x \), then \( \frac{1}{\sqrt{3}} \, f^{\prime}\!\left(\frac{1}{\sqrt{3}}\right) \) is equal to

The integral $\int_{0}^{1} \frac{1}{7^{\left[\frac{1}{x}\right]}} dx$, where $[\,.\,]$ denotes the greatest integer function, is equal to

If \( n(2n+1)\displaystyle\int\limits_{0}^{1}(1-x^{n})^{2n}\,dx=1177\int\limits_{0}^{1}(1-x^{n})^{2n+1}\,dx \), then \( n\in\mathbb{N} \) is equal to ___________.

The value of the integral \( \frac{48}{\pi^{4}} \int_{0}^{{\pi}{}} \left(\frac{3\pi x^{2}}{2}-x^{3}\right)^{}\frac{\sin x}{1+\cos^{2}x}\,dx \) is equal to ____________ .

Let \( I=\displaystyle \int\limits_{\pi/4}^{\pi/3}\left(\frac{8\sin x-\sin 2x}{x}\right)\,dx \). Then

The value of integral \(\displaystyle \int\limits_{0}^{\pi/2} 60\,\frac{\sin(6x)}{\sin x}\,dx \) is equal to __________________.

The area of the region enclosed between the parabolas \( y^{2} = 2x - 1 \) and \( y^{2} = 4x - 3 \) is

The area of the region enclosed by \( y \le 4x^{2} \), \( x^{2} \le 9y \) and \( y \le 4 \), is equal to:

Let $\displaystyle \max_{0 \le x \le 2} \left\{ \frac{9-x^2}{5-x} \right\} = \alpha$ and $\displaystyle \min_{0 \le x \le 2} \left\{ \frac{9-x^2}{5-x} \right\} = \beta$.
If $\int_{\beta - \frac{8}{3}}^{2\alpha - 1} max \left\{ \frac{9-x^2}{5-x}, x \right\} dx = \alpha_1 + \alpha_2 \log_e \left( \frac{8}{15} \right)$ then $\alpha_1 + \alpha_2$ is equal to __________.

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 \( 3 \) and \( x(>3) \) be \( \left(\dfrac{y}{x}\right)^{3} \). If this curve also passes through the point \( (\alpha,6\sqrt{10}) \) in the first quadrant, then \( \alpha \) is equal to ______.

If \( \int_{0}^{2}\left(\sqrt{2x}-\sqrt{2x-x^{2}}\right)dx=\int_{0}^{1}\left(1-\sqrt{1-y^{2}}-\frac{y^{2}}{2}\right)dy+\int_{1}^{2}\left(2-\frac{y^{2}}{2}\right)dy+I \) then \( I \) equals:

The value of the integral \( \displaystyle \int_{-\pi/2}^{\pi/2} \frac{dx}{(1+e^{x})(\sin^{6}x+\cos^{6}x)} \) is equal to:

\( \displaystyle \int\limits_{0}^{20\pi} (|\sin x|+|\cos x|)^2 \, dx \) is equal to

Let \( f(\theta) = \sin \theta + \displaystyle \int_{-\pi/2}^{\pi/2} (\sin \theta + t \cos \theta)\, f(t)\, dt \). Then the value of \( \left| \displaystyle \int_{0}^{\pi/2} f(\theta)\, d\theta \right| \) is ____________.

The odd natural number \( a \), such that the area of the region bounded by \( y=1 \), \( y=3 \), \( x=0 \), \( x=y^{a} \) is \( \dfrac{364}{3} \), is equal to:

The area of the region \( \{(x,y): |x-1| \le y \le \sqrt{5-x^2}\} \) is equal to:

Let \( A_1 = \{(x,y): |x| \le y^{2}, |x| + 2y \le 8\} \) and \( A_2 = \{(x,y): |x| + |y| \le k\} \). If \( 27(\text{Area } A_1) = 5(\text{Area } A_2) \) then \( k \) is equal to:

The area of the region \( S = \{(x,y) : y^2 \le 8x,\, y \ge \sqrt{2}x,\, x \ge 1\} \) is:

Let \( S \) be the region bounded by the curves \( y = x^{3} \) and \( y^{2} = x \). The curve \( y = 2|x| \) divides \( S \) into two regions of areas \( R_{1} \) and \( R_{2} \).
If \( \max\{R_{1}, R_{2}\} = R_{2} \), then \( \dfrac{R_{2}}{R_{1}} \) is equal to ______________.

Let \( f : R \to R \) be a differentiable function such that \( f\!\left(\frac{\pi}{4}\right) = \sqrt{2} , \ f\!\left(\frac{\pi}{2}\right) = 0 \) and \( f'\!\left(\frac{\pi}{2}\right) = 1 \) and let
\( g(x) = \int_{x}^{\pi/4} \left( f'(t)\sec t + \tan t \sec t \ f(t) \right) dt \) for \( x \in \left[ \frac{\pi}{4}, \frac{\pi}{2} \right) \). Then \( \displaystyle \lim_{x \to \left(\frac{\pi}{2}\right)^{-}} g(x) \) is equal to

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JEE Main 2022 24th June Morning Shift

If \( x = x(y) \) is the solution of the differential equation \( y\,\frac{dx}{dy} = 2x + y^3 (y+1) e^{y} \), \( x(1) = 0 \); then \( x(e) \) is equal to: