JEE Main 2024 — Calculus

Let a rectangle $A B C D$ of sides 2 and 4 be inscribed in another rectangle $P Q R S$ such that the vertices of the rectangle $A B C D$ lie on the sides of the rectangle $P Q R S$. Let $a$ and $b$ be the sides of the rectangle $P Q R S$ when its area is maximum. Then $(a+b)^2$ is equal to :

The derivative of ln(sin x) with respect to x is:

The derivative of sin²x is

The integral of 1/x is

If the integral from 0 to 1 of x²eˣ dx = ae - b, find a + b.

The derivative of sin(x) with respect to x is:

If $f(x)=\left\{\begin{array}{l}x^3 \sin \left(\frac{1}{x}\right), x \neq 0 \\ 0 \quad, x=0\end{array}\right.$ then

Let $f(x)=3 \sqrt{x-2}+\sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha^2+2 \beta^2$ is equal to

Consider the function $f(x)={\begin{matrix}\frac{a(7x-12-{x}^{2})}{b|{x}^{2}-7x+12|} & ,x<3 \\ {2}^{\frac{\mathrm{sin}(x-3)}{x-[x]}} & ,x>3 \\ b & ,x=3\end{matrix}$,where $[x]$ denotes the greatest integer less than or equal to $x$. If $S$ denotes the set of all ordered pairs $(a,b)$ such that $f(x)$is continuous at $x=3$, then the number of elements in $S$ is :

Consider the function $f:(0,2)\rightarrow R$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by $g(x)={\begin{matrix}\text{min}{f(t)}, & 0<t\leq x\text{ and }0<x\leq 1 \\ \frac{3}{2}+x, & 1<x<2\end{matrix}$. Then

If $\mathrm{sin}(\frac{y}{x})={\mathrm{log}}_{e}|x|+\frac{\alpha }{2}$ is the solution of the differential equation $x\mathrm{cos}(\frac{y}{x})\frac{dy}{dx}=y\mathrm{cos}(\frac{y}{x})+x$ and $y(1)=\frac{\pi }{3}$, then ${\alpha }^{2}$ is equal to

Let $y=y(x)$ be the solution of the differential equation $\left(2 x \log _e x\right) \frac{d y}{d x}+2 y=\frac{3}{x} \log _e x, x>0$ and $y\left(e^{-1}\right)=0$. Then, $y(e)$ is equal to

The number of critical points of the function $f(x)=(x-2)^{2 / 3}(2 x+1)$ is

$\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right)$ is equal to

If $\int \frac{1}{\mathrm{a}^2 \sin ^2 x+\mathrm{b}^2 \cos ^2 x} \mathrm{~d} x=\frac{1}{12} \tan ^{-1}(3 \tan x)+$ constant, then the maximum value of $\mathrm{a} \sin x+\mathrm{b} \cos x$, is :

If ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\sqrt{1-\mathrm{sin}2x}dx=\alpha +\beta \sqrt{2}+\gamma \sqrt{3}$, where $\alpha ,\beta$ and $\gamma$ are rational numbers, then $3\alpha +4\beta -\gamma$ is equal to _____.

The parabola $y^2=4 x$ divides the area of the circle $x^2+y^2=5$ in two parts. The area of the smaller part is equal to:

Let $\alpha$ be a non-zero real number. Suppose $f:R\rightarrow R$ is a differentiable function such that $f(0)=1$ and $\underset{x\rightarrow -\infty }{\mathrm{lim}}f(x)=1$. If ${f}^{'}(x)=\alpha f(x)+3$, for all $x\in R,$ then $f(-{\mathrm{log}}_{e}2)$ is equal to ________.

Let $\beta(\mathrm{m}, \mathrm{n})=\int_0^1 x^{\mathrm{m}-1}(1-x)^{\mathrm{n}-1} \mathrm{~d} x, \mathrm{~m}, \mathrm{n}>0$. If $\int_0^1\left(1-x^{10}\right)^{20} \mathrm{~d} x=\mathrm{a} \times \beta(\mathrm{b}, \mathrm{c})$, then $100(\mathrm{a}+\mathrm{b}+\mathrm{c})$ equals____

If $\int _{0}^{\frac{\pi }{3}}{\mathrm{cos}}^{4}xdx=a\pi +b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a+8b$ is equal to:

Let $\int_\alpha^{\log _e 4} \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=\frac{\pi}{6}$. Then $\mathrm{e}^\alpha$ and $\mathrm{e}^{-\alpha}$ are the roots of the equation :

For $0<a<1$, the value of the integral ${\int }_{0}^{\pi }\frac{dx}{1-2a\mathrm{cos}x+{a}^{2}}$ is :

Let the solution $y=y(x)$ of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $y\left(\frac{\pi}{2}\right)+10$ is equal to ______

$\lim _{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}$ is equal to

The value of $\lim _{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots \ldots \sqrt[10]{\cos 10 x}}{x^2}\right)$ is

$\lim _{n \rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\cdots+\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\cdots \cdots+n^3\right)-\left(1^2+2^2+\cdots \cdots+n^2\right)}$ is equal to :

Let $f(x)={\begin{matrix}x-1,x\text{is even}, \\ 2x,x\text{is odd},\end{matrix}x\in N$. If for some $a\in N,f(f(f(a)))=21$, then $\underset{x\rightarrow {a}^{-}}{\mathrm{lim}}{\frac{{|x|}^{3}}{a}-[\frac{x}{a}]},$ where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to:

If $\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{\mathrm{m} \sqrt{5}}{\mathrm{n}(2 \mathrm{n})^{2 / 3}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $8 \mathrm{~m}+12 \mathrm{n}$ is equal to______

Let $a$ be the sum of all coefficients in the expansion of $(1–2x+2{x}^{2}{)}^{2023}(3-4{x}^{2}+2{x}^{3}{)}^{2024}$ and $b=\underset{x\rightarrow 0}{\mathrm{lim}}(\frac{{\int }_{0}^{x}\frac{\mathrm{log}(1+t)}{{t}^{2024}+1}dt}{{x}^{2}})$. If the equations $c{x}^{2}+dx+e=0$ and $2b{x}^{2}+ax+4=0$ have a common root, where $c,d,e\in R$, then $d:c:e$ equals

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{a{x}^{2}{e}^{x}-b{\mathrm{log}}_{e}(1+x)+cx{e}^{-x}}{{x}^{2}\mathrm{sin}x}=1$, then $16({a}^{2}+{b}^{2}+{c}^{2})$ is equal to ______.

If $a=\underset{x\rightarrow 0}{\mathrm{lim}}\frac{\sqrt{1+\sqrt{1+{x}^{4}}}-\sqrt{2}}{{x}^{4}}$ and $b=\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\mathrm{sin}}^{2}x}{\sqrt{2}-\sqrt{1+\mathrm{cos}x}}$, then the value of $a{b}^{3}$ is :

Let the slope of the line $45x+5y+3=0$ be $27{r}_{1}+\frac{9{r}_{2}}{2}$ for some ${r}_{1},{r}_{2}\in R$. Then $\underset{x\rightarrow 3}{\mathrm{lim}}({\int }_{3}^{x}\frac{8{t}^{2}}{\frac{3{r}_{2}x}{2}-{r}_{2}{x}^{2}-{r}_{1}{x}^{3}-3x}dt)$ is equal to ______.

The solution curve, of the differential equation $2 y \frac{\mathrm{d} y}{\mathrm{~d} x}+3=5 \frac{\mathrm{d} y}{\mathrm{~d} x}$, passing through the point $(0,1)$ is a conic, whose vertex lies on the line:

Three points $O(0,0),P(a,{a}^{2}),Q(-b,{b}^{2}),a>0,b>0,$ are on the parabola $y={x}^{2}$. Let ${S}_{1}$ be the area of the region bounded by the line $PQ$ and the parabola, and ${S}_{2}$ be the area of the triangle $OPQ$. If the minimum value of $\frac{{S}_{1}}{{S}_{2}}$ is $\frac{m}{n},\mathrm{gcd}(m,n)=1,$ then $m+n$ is equal to:

Let $f:(-\infty, \infty)-\{0\} \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(1)=\lim _{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$. Then $\lim _{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^2-2 \log _e a$ is equal to

If the solution $y=y(x)$ of the differential equation $\left(x^4+2 x^3+3 x^2+2 x+2\right) \mathrm{d} y-\left(2 x^2+2 x+3\right) \mathrm{d} x=0$ satisfies $y(-1)=-\frac{\pi}{4}$, then $y(0)$ is equal to :

Let $y=y(x)$ be the solution of the differential equation $(x+y+2)^2 d x=d y, y(0)=-2$. Let the maximum and minimum values of the function $y=y(x)$ in $\left[0, \frac{\pi}{3}\right]$ be $\alpha$ and $\beta$, respectively. If $(3 \alpha+\pi)^2+\beta^2=\gamma+\delta \sqrt{3}, \gamma, \delta \in \mathbb{Z}$, then $\gamma+\delta$ equals ______

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{e}^{2|\mathrm{sin}x|}-2|\mathrm{sin}x|-1}{{x}^{2}}$

Let $f:[-\frac{\pi }{2},\frac{\pi }{2}]\rightarrow R$ be a differentiable function such that $f(0)=\frac{1}{2}$, If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{x{\int }_{0}^{x}f(t)dt}{{e}^{{x}^{2}}-1}=\alpha$, then $8{\alpha }^{2}$ is equal to :

Let $\mathrm{A}$ be the region enclosed by the parabola $y^2=2 x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $\mathrm{A}$ is________

Let the area of the region ${(x,y):0\leq x\leq 3,0\leq y\leq$ $\mathrm{min}{{x}^{2}+2,2x+2}}$ be $A$. Then $12A$ is equal to ______.

Let $f(x)=|2{x}^{2}+5|x|-3|,x\in R$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m+n$ is equal to:

Let $\mathrm{a}>0$ be a root of the equation $2 x^2+x-2=0$. If $\lim _{x \rightarrow \frac{1}{\mathrm{a}}} \frac{16\left(1-\cos \left(2+x-2 x^2\right)\right)}{(1-\mathrm{a} x)^2}=\alpha+\beta \sqrt{17}$, where $\alpha, \beta \in Z$, then $\alpha+\beta$ is equal to_______

A function $y=f(x)$ satisfies $f(x)\mathrm{sin}2x+\mathrm{sin}x-(1+{\mathrm{cos}}^{2}x){f}^{'}(x)=0$ with condition $f(0)=0$. Then $f(\frac{\pi }{2})$ is equal to

Let $f(x)=\int_0^x\left(t+\sin \left(1-e^{\prime}\right)\right) d t, x \in \mathbb{R}$. Then, $\lim _{x \rightarrow 0} \frac{f(x)}{x^3}$ is equal to

Let $y={\mathrm{log}}_{e}(\frac{1-{x}^{2}}{1+{x}^{2}}),-1<x<1$. Then at $x=\frac{1}{2}$, the value of $225({y}^{'}-{y}^{"})$ is equal to

If ${\int }_{0}^{1}\frac{1}{\sqrt{3+x}+\sqrt{1+x}}dx=a+b\sqrt{2}+c\sqrt{3}$, where $a,b,c$ are rational numbers, then $2a+3b-4c$ is equal to :

If $5f(x)+4f(\frac{1}{x})={x}^{2}-2,\forall x\neq 0$ and $y=9{x}^{2}f(x),$ then $y$ is strictly increasing in:

If $\frac{dx}{dy}=\frac{1+x-{y}^{2}}{y},x(1)=1,$ then $5x(2)$ is equal to:

For $\mathrm{a}, \mathrm{b}>0$, let $f(x)=\left\{\begin{array}{cc}\frac{\tan ((\mathrm{a}+1) x)+\mathrm{b} \tan x}{x}, & x < 0 \\ 3, & x=0 \\ \frac{\sqrt{\mathrm{a} x+\mathrm{b}^2 x^2}-\sqrt{\mathrm{a} x}}{\mathrm{~b} \sqrt{\mathrm{a}} x \sqrt{x}}, & x>0\end{array}\right.$ be a continous function at $x=0$. Then $\frac{\mathrm{b}}{\mathrm{a}}$ is equal to :

Let $f:(0, \pi) \rightarrow \mathbf{R}$ be a function given by $f(x)=$ \(\left\{\begin{array}{cc}\left(\frac{8}{7}\right)^{\frac{\tan 8 x}{\tan 7 x}}, & 0 < x < \frac{\pi}{2} \\ \mathrm{a}-8, & x=\frac{\pi}{2} \\ (1+|\cot x|)^{\frac{\mathrm{b}}{}|\tan x|}, & \frac{\pi}{2} < x < \pi\end{array}\right.\) where $\mathrm{a}, \mathrm{b} \in \mathbf{Z}$. If $f$ is continuous at $x=\frac{\pi}{2}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to

Let $[\mathrm{t}]$ denote the greatest integer less than or equal to $\mathrm{t}$. Let $f:[0, \infty) \rightarrow \mathbf{R}$ be a function defined by $f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]$. Let $S$ be the set of all points in the interval $[0,8]$ at which $f$ is not continuous. Then $\sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to _______

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function given by $f(x)=\left\{\begin{array}{ll} \frac{1-\cos 2 x}{x^2}, & x < 0 \\ \alpha, & x=0, \\ \frac{\beta \sqrt{1-\cos x}}{x}, & x>0 \end{array}\right.$ where $\alpha, \beta \in \mathbf{R}$. If $f$ is continuous at $x=0$, then $\alpha^2+\beta^2$ is equal to :

If the function $f(x)=\frac{\sin 3 x+\alpha \sin x-\beta \cos 3 x}{x^3}, x \in \mathbf{R}$, is continuous at $x=0$, then $f(0)$ is equal to :

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}\frac{a-b\mathrm{cos}2x}{{x}^{2}};x<0 \\ {x}^{2}+cx+2;0\leq x\leq 1 \\ 2x+1;x>1\end{matrix}$ If $f$ is continuous everywhere in $R$ and $m$ is the number of points where $f$ is NOT differential then $m+a+b+c$ equals:

Let $f:R-{0}\rightarrow R$ be a function satisfying $f(\frac{x}{y})=\frac{f(x)}{f(y)}$ for all $x,y,f(y)\neq 0$. If ${f}^{'}(1)=2024$, then

Consider the function $f:(0,\infty )\rightarrow R$ defined by $f(x)={e}^{-|{\mathrm{log}}_{e}x|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m+n$ is

If the function $f(x)={\begin{matrix}\frac{1}{|x|}, & |x|\geq 2 \\ a{x}^{2}+2b, & |x|<2\end{matrix}$ is differentiable on $R$, then $48(a+b)$ is equal to _______.

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)={\begin{matrix}{x}^{2}+3x+a, & x\leq 1 \\ bx+2, & x>1\end{matrix}$ be differentiable on $R$. Then, the value of ${\int }_{-2}^{2}f(x)dx$ equals

Let $f(x)=\sqrt{\underset{r\rightarrow x}{\mathrm{lim}}{\frac{2{r}^{2}[(f(r){)}^{2}-f(x)f(r)]}{{r}^{2}-{x}^{2}}-{r}^{3}{e}^{\frac{f(r)}{r}}}}$ be differentiable in $(-\infty ,0)\cup (0,\infty )$ and $f(1)=1$. Then the value of $ae$, such that $f(a)=0$, is equal to ______.

Suppose for a differentiable function $h, h(0)=0, h(1)=1$ and $h^{\prime}(0)=h^{\prime}(1)=2$. If $\mathrm{g}(x)=h\left(\mathrm{e}^x\right) \mathrm{e}^{h(x)}$, then $g^{\prime}(0)$ is equal to:

If $y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}$, then at $\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y$ is equal to :

Suppose $f(x)=\frac{({2}^{x}+{2}^{-x})\mathrm{tan}x\sqrt{{\mathrm{tan}}^{-1}({x}^{2}-x+1)}}{{(7{x}^{2}+3x+1)}^{3}}$. Then the value of ${f}^{'}(0)$ is equal to

Let $f(x)={x}^{3}+{x}^{2}{f}^{'}(1)+x{f}^{"}(2)+{f}^{'''}(3),x\in R$. Then ${f}^{'}(10)$ is equal to

Let $f(x)=x^5+2 \mathrm{e}^{x / 4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8 g^{\prime}(2)$ is :

Let for a differentiable function $f:(0,\infty )\rightarrow R$, $f(x)-f(y)\geq {\mathrm{log}}_{e}(\frac{x}{y})+x-y,\forall x,y\in (0,\infty )$. Then $\sum _{n=1}^{20}{f}^{'}(\frac{1}{{n}^{2}})$ is equal to

Let the set of all values of $p$, for which $f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7$ does not have any critical point, be the interval $(a, b)$. Then $16 a b$ is equal to _______

Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $\left(1+x\left(\lambda^2-x^2\right)\right)$ satisfies $\frac{x^2+x+2}{x^2+5 x+6} < 0$, be $(\alpha, \beta)$. Then $\alpha^2+\beta^2$ is equal to _________

A variable line $L$ passes through the point $(3,5)$ and intersects the positive coordinate axes at the points $\mathrm{A}$ and $\mathrm{B}$. The minimum area of the triangle $\mathrm{OAB}$, where $\mathrm{O}$ is the origin, is :

For the function $f(x)=(\cos x)-x+1, x \in \mathbb{R}$, between the following two statements (S1) $f(x)=0$ for only one value of $x$ in $[0, \pi]$. (S2) $f(x)$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$.

Let $f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10$. The number of points of local maxima of $f$ in interval $(0,2 \pi)$ is

The interval in which the function $f(x)=x^x, x>0$, is strictly increasing is

Let the maximum and minimum values of $\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in \mathbf{R}$ be $\mathrm{M}$ and $\mathrm{m}$, respectively. Then $\mathrm{M}^2-\mathrm{m}^2$ is equal to _________

For the function $f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$ consider the following two statements : (I) $\mathrm{f}$ is increasing in $\left(0, \frac{\pi}{2}\right)$. (II) $f^{\prime}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$. Between the above two statements,

Let $f(x)={(x+3)}^{2}{(x-2)}^{3},x\in [-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4,4]$, then the value of $M-m$ is :

The function $f(x)=\frac{x}{{x}^{2}-6x-16},x\in \mathbb{R}-{-2,8}$

If the integral $525\int _{0}^{\frac{\pi }{2}}\mathrm{sin}2x{\mathrm{cos}}^{\frac{11}{2}}x{(1+{\mathrm{cos}}^{\frac{5}{2}}x)}^{\frac{1}{2}}dx$ is equal to $(n\sqrt{2}-64)$, then $n$ is equal to ________

Let $\int \frac{2-\tan x}{3+\tan x} \mathrm{~d} x=\frac{1}{2}\left(\alpha x+\log _{\mathrm{e}}|\beta \sin x+\gamma \cos x|\right)+C$, where $C$ is the constant of integration. Then $\alpha+\frac{\gamma}{\beta}$ is equal to :

If $\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} \mathrm{~d} x=\mathrm{A}\left(\frac{\alpha x-1}{\beta x+3}\right)^B+\mathrm{C}$, where $\mathrm{C}$ is the constant of integration, then the value of $\alpha+\beta+20 \mathrm{AB}$ is__________

Let $I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x$. If $I(0)=3$, then $I\left(\frac{\pi}{12}\right)$ is equal to

Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f(1)=1$ and $\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x>0$. Then $2 f(2)+3 f(3)$ is equal to _______

The value of the integral $\int_{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x$ is

Let $\int_0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} d t=\int_0^x y(t) d t, 0 \leq x \leq 3, y \geq 0, y(0)=0$. Then at $x=2, y^{\prime \prime}+y+1$ is equal to

Let $[t]$ denote the largest integer less than or equal to $t$. If $\int_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) \mathrm{d} x=\mathrm{a}+\mathrm{b} \sqrt{2}-\sqrt{3}-\sqrt{5}+\mathrm{c} \sqrt{6}-\sqrt{7}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbf{Z}$, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to_______

$\int_0^{\pi / 4} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x$ is equal to

The value of $k \in \mathrm{N}$ for which the integral $I_n=\int_0^1\left(1-x^k\right)^n d x, n \in \mathbb{N}$, satisfies $147 I_{20}=148 I_{21}$ is

Let $r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in \mathbb{N}$. Then the value of $\sum_{k=1}^{10} \frac{1}{7\left(r_k-1\right)}$ is equal to________

If the value of the integral $\int_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x$ is $\frac{2}{\pi}$. Then, a value of $\alpha$ is

The integral $\int_0^{\pi / 4} \frac{136 \sin x}{3 \sin x+5 \cos x} d x$ is equal to :

If $\int_0^{\frac{\pi}{4}} \frac{\sin ^2 x}{1+\sin x \cos x} \mathrm{~d} x=\frac{1}{\mathrm{a}} \log _{\mathrm{e}}\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi}{\mathrm{b} \sqrt{3}}$, where $\mathrm{a}, \mathrm{b} \in \mathbf{N}$, then $\mathrm{a}+\mathrm{b}$ is equal to_________

If ${\int }_{-\pi /2}^{\pi /2}\frac{8\sqrt{2}\mathrm{cos}xdx}{(1+{e}^{\mathrm{sin}x})(1+{\mathrm{sin}}^{4}x)}=\alpha \pi +\beta {\mathrm{log}}_{e}(3+2\sqrt{2})$, where $\alpha ,\beta$ are integers, then ${\alpha }^{2}+{\beta }^{2}$ equals __________

Let $f(x)=\left\{\begin{array}{lr}-2, & -2 \leq x \leq 0 \\ x-2, & 0 < x \leq 2\end{array}\right.$ and $h(x)=f(|x|)+|f(x)|$. Then $\int_{-2}^2 h(x) \mathrm{d} x$ is equal to :

The value of ${\int }_{0}^{1}{(2{x}^{3}-3{x}^{2}-x+1)}^{\frac{1}{3}}dx$ is equal to:

Let $f:(0,\infty )\rightarrow R$ and $F(x)=\int _{0}^{x}tf(t)dt$. If $F({x}^{2})={x}^{4}+{x}^{5},$ then $\sum _{r=1}^{12}f({r}^{2})$ is equal to:

$|\frac{120}{{\pi }^{3}}\int _{0}^{\pi }\frac{{x}^{2}\mathrm{sin}x\mathrm{cos}x}{{\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x}dx|$ is equal to ______.

The value of the integral $\int _{0}^{\frac{\pi }{4}}\frac{xdx}{{\mathrm{sin}}^{4}(2x)+{\mathrm{cos}}^{4}(2x)}$ equals:

Let $S=(-1,\infty )$ and $f:S\rightarrow \mathbb{R}$ be defined as $f(x)=\int _{-1}^{x}{({e}^{t}-1)}^{11}{(2t-1)}^{5}{(t-2)}^{7}{(t-3)}^{12}{(2t-10)}^{61}dt$. Let $p=$ Sum of square of the values of $x$, where $f(x)$ attains local maxima on $S$. and $q=$Sum of the values of $x$, where $f(x)$ attains local minima on $S$. Then, the value of ${p}^{2}+2q$ is ________

Let $f:R\rightarrow R$ be defined $f(x)=a{e}^{2x}+b{e}^{x}+cx$. If $f(0)=-1,{f}^{'}({\mathrm{log}}_{e}2)=21$ and ${\int }_{0}^{\mathrm{log}4}(f(x)-cx)dx=\frac{39}{2}$, then the value of $|a+b+c|$ equals:

The value $9{\int }_{0}^{9}[\sqrt{\frac{10x}{x+1}}]\mathrm{dx}$, where $t$ denotes the greatest integer less than or equal to $t$, is _____.

Let $f(x)={\int }_{0}^{x}g(t){\mathrm{log}}_{e}(\frac{1-t}{1+t})\mathrm{dt}$, where $g$ is a continuous odd function. If ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(f(x)+\frac{{x}^{2}\mathrm{cosx}}{1+{e}^{x}})\mathrm{dx}={(\frac{\pi }{\alpha })}^{2}-\alpha$, then $\alpha$ is equal to _____.

For $x\in (-\frac{\pi }{2},\frac{\pi }{2})$, if $y(x)=\int \frac{cosecx+\mathrm{sin}x}{cosecx\mathrm{sec}x+\mathrm{tan}x{\mathrm{sin}}^{2}x}dx$ and $\underset{x\rightarrow {(\frac{\pi }{2})}^{-}}{\mathrm{lim}}y(x)=0$ then $y(\frac{\pi }{4})$ is equal to

If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and ${I}_{1}={\int }_{a}^{b}\mathrm{xsin}(4x-{x}^{2})\mathrm{dx},{I}_{2}={\int }_{a}^{b}\mathrm{sin}(4x-{x}^{2})\mathrm{dx}$ , then $36\frac{{I}_{1}}{{I}_{2}}$ is equal to :

Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{ax}=0$ and $\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm{by}=0$ respectively, $a,b\in R$. Given that $x(0)=2;y(0)=1$ and $3y(1)=2x(1)$, the value of $t$, for which $x(t)=y(t)$, is :

Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1,f(1))$ and $(3,f(3))$ make angles $\frac{\pi }{6}$ and $\frac{\pi }{4}$, respectively with positive $x$-axis. If $27{\int }_{1}^{3}({({f}^{'}(t))}^{2}+1){f}^{"}(t)dt=\alpha +\beta \sqrt{3}$ where $\alpha ,\beta$ are integers, then the value of $\alpha +\beta$ equals

The area (in square units) of the region enclosed by the ellipse $x^2+3 y^2=18$ in the first quadrant below the line $y=x$ is

The area of the region in the first quadrant inside the circle $x^2+y^2=8$ and outside the parabola $y^2=2 x$ is equal to :

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4 a^2+a-1$. Then the differential equation, whose general solution is $y=c_1 f(x)+c_2$, where $c_1$ and $c_2$ are arbitrary constants, is

Let the area of the region enclosed by the curve $y=\min \{\sin x, \cos x\}$ and the $x$ axis between $x=-\pi$ to $x=\pi$ be $A$. Then $A^2$ is equal to ___________

The area (in sq. units) of the region described by $\left\{(x, y): y^2 \leq 2 x\right.$, and $\left.y \geq 4 x-1\right\}$ is

The area enclosed between the curves $y=x|x|$ and $y=x-|x|$ is :

The area enclosed by the curves $xy+4y=16$ and $x+y=6$ is equal to:

The area of the region ${(x,y):{y}^{2}\leq 4x,x<4,\frac{xy(x-1)(x-2)}{(x-3)(x-4)}>0,x\neq 3}$ is

The area of the region enclosed by the parabola $(y-2{)}^{2}=x-1$, the line $x-2y+4=0$ and the positive coordinate axes is __________.

If the area of the region ${(x,y):0\leq y\leq \mathrm{min}{2x,6x-{x}^{2}}}$ is $A$, then $12A$ is equal to _______.

Let the area of the region ${(x,y):x-2y+4\geq 0$, $x+2{y}^{2}\geq 0,x+4{y}^{2}\leq 8,y\geq 0}$ be $\frac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $m+n$ is equal to ______.

The area (in sq. units) of the part of circle ${x}^{2}+{y}^{2}=169$ which is below the line $5x-y=13$ is $\frac{\pi \alpha }{2\beta }-\frac{65}{2}+\frac{\alpha }{\beta }{\mathrm{sin}}^{-1}(\frac{12}{13})$ where $\alpha ,\beta$ are coprime numbers. Then $\alpha +\beta$ is equal to

The solution of the differential equation $\left(x^2+y^2\right) \mathrm{d} x-5 x y \mathrm{~d} y=0, y(1)=0$, is :

Suppose the solution of the differential equation $\frac{d y}{d x}=\frac{(2+\alpha) x-\beta y+2}{\beta x-2 \alpha y-(\beta \gamma-4 \alpha)}$ represents a circle passing through origin. Then the radius of this circle is :

Let $y=y(x)$ be the solution of the differential equation $\left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \tan x}\right) d y=0, y(0)=1$. Then $y\left(\frac{\pi}{4}\right)$ is equal to

If the solution $y(x)$ of the given differential equation $\left(\mathrm{e}^y+1\right) \cos x \mathrm{~d} x+\mathrm{e}^y \sin x \mathrm{~d} y=0$ passes through the point $\left(\frac{\pi}{2}, 0\right)$, then the value of $\mathrm{e}^{y\left(\frac{\pi}{6}\right)}$ is equal to_________

If $y=y(x)$ is the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=\sin (2 x), y(0)=\frac{3}{4}$, then $y\left(\frac{\pi}{8}\right)$ is equal to:

If $x=x(t)$ is the solution of the differential equation $(t+1)dx=(2x+{(t+1)}^{4})dt,x(0)=2$, then $x(1)$ equals ________

Let $y=y(x)$ be the solution of the differential equation ${\mathrm{sec}}^{2}xdx+({e}^{2y}{\mathrm{tan}}^{2}x+\mathrm{tan}x)dy=0,$ $0<x<\frac{\pi }{2},y(\frac{\pi }{4})=0$. If $y(\frac{\pi }{6})=\alpha$, then ${e}^{8\alpha }$ is equal to ______.

The solution curve of the differential equation $y\frac{dx}{dy}=x({\mathrm{log}}_{e}x-{\mathrm{log}}_{e}y+1),x>0,y>0$ passing through the point $(e,1)$ is

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{(\mathrm{tan}x)+y}{\mathrm{sin}x(\mathrm{sec}x-\mathrm{sin}x\mathrm{tan}x)},x\in (0,\frac{\pi }{2})$ satisfying the condition $y(\frac{\pi }{4})=2$. Then, $y(\frac{\pi }{3})$ is

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=2x{(x+y)}^{3}-x(x+y)-1,y(0)=1$. Then, ${(\frac{1}{\sqrt{2}}+y(\frac{1}{\sqrt{2}}))}^{2}$ equals:

Let $y=y(x)$ be the solution of the differential equation $\mathrm{sec}xdy+{2(1-x)\mathrm{tan}x+x(2-x)}dx=0$ such that $y(0)=2$. Then $y(2)$ is equal to :

If the solution curve $y=y(x)$ of the differential equation $(1+{y}^{2})(1+{\mathrm{log}}_{e}x)dx+xdy=0,x>0$ passes through the point $(1,1)$ and $y(e)=\frac{\alpha -\mathrm{tan}(\frac{3}{2})}{\beta +\mathrm{tan}(\frac{3}{2})}$, then $\alpha +2\beta$ is

If the solution curve, of the differential equation $\frac{dy}{dx}=\frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is ${\mathrm{tan}}^{-1}(\frac{y-1}{x-1})-\frac{1}{\beta }{\mathrm{log}}_{e}(\alpha +{(\frac{y-1}{x-1})}^{2})={\mathrm{log}}_{e}|x-1|$, then $5\beta +\alpha$ is equal to

If the solution of the differential equation $(2x+3y-2)dx+(4x+6y-7)dy=0,y(0)=3$, is $\alpha x+\beta y+3{\mathrm{log}}_{e}|2x+3y-\gamma |=6$, then $\alpha +2\beta +3\gamma$ is equal to ______.

$\underset{x\rightarrow \frac{\pi }{2}}{\mathrm{lim}}(\frac{1}{{(x-\frac{\pi }{2})}^{2}}{\int }_{{x}^{3}}^{{(\frac{\pi }{2})}^{3}}\mathrm{cos}(\frac{1}{{t}^{3}})dt)$ is equal to

Let $f(x)=a x^3+b x^2+c x+41$ be such that $f(1)=40, f^{\prime}(1)=2$ and $f^{\prime}(1)=4$. Then $\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2$ is equal to:

The value of $\int_{-\pi}^\pi \frac{2 y(1+\sin y)}{1+\cos ^2 y} d y$ is :

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+4\right)^2 d y+\left(2 x^3 y+8 x y-2\right) d x=0$. If $y(0)=0$, then $y(2)$ is equal to

If $\log _e y=3 \sin ^{-1} x$, then $\left(1-x^2\right) y^{\prime \prime}-x y^{\prime}$ at $x=\frac{1}{2}$ is equal to

If $\alpha=\lim _{x \rightarrow 0^{+}}\left(\frac{\mathrm{e}^{\sqrt{\tan x}}-\mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$ and $\beta=\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}$ are the roots of the quadratic equation $a x^2+b x-\sqrt{\mathrm{e}}=0$, then $12 \log _{\mathrm{e}}(\mathrm{a}+\mathrm{b})$ is equal to__________

If $y=\frac{(\sqrt{x}+1)({x}^{2}-\sqrt{x})}{x\sqrt{x}+x+\sqrt{x}}+\frac{1}{15}(3{\mathrm{cos}}^{2}x-5){\mathrm{cos}}^{3}x,$ then $96{y}^{'}(\frac{\pi }{6})$ is equal to:

Let $f:[-1,2] \rightarrow \mathbf{R}$ be given by $f(x)=2 x^2+x+\left[x^2\right]-[x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{3+\alpha \mathrm{sin}x+\beta \mathrm{cos}x+{\mathrm{log}}_{e}(1-x)}{3{\mathrm{tan}}^{2}x}=\frac{1}{3}$, then $2\alpha -\beta$ is equal to :

The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2{y}^{2}=kx$ and $k{y}^{2}=2(y-x)$ is maximum, is equal to:

Let $g(x)=3f(\frac{x}{3})+f(3-x)$ and ${f}^{"}(x)>0$ for all $x\in (0,3)$. If $g$ is decreasing in $(0,\alpha )$ and increasing in $(\alpha ,3)$, then $8\alpha$ is

If the area of the region $\left\{(x, y): \frac{\mathrm{a}}{x^2} \leq y \leq \frac{1}{x}, 1 \leq x \leq 2,0 < \mathrm{a} < 1\right\}$ is $\left(\log _{\mathrm{e}} 2\right)-\frac{1}{7}$ then the value of $7 \mathrm{a}-3$ is equal to:

If $y=y(x)$ is the solution curve of the differential equation $({x}^{2}-4)\mathrm{dy}-({y}^{2}-3y)\mathrm{dx}=0$, $x>2,y(4)=\frac{3}{2}$ and the slope of the curve is never zero, then the value of $y(10)$ equals :

If $f(t)=\int_0^\pi \frac{2 x \mathrm{~d} x}{1-\cos ^2 \mathrm{t} \sin ^2 x}, 0 < \mathrm{t} < \pi$, then the value of $\int_0^{\frac{\pi}{2}} \frac{\pi^2 \mathrm{dt}}{f(\mathrm{t})}$ equals_________

If the function $f(x)=\left(\frac{1}{x}\right)^{2 x} ; x>0$ attains the maximum value at $x=\frac{1}{\mathrm{e}}$ then :

Let $y=y(x)$ be the solution of the differential equation $(1-{x}^{2})dy=[xy+({x}^{3}+2)\sqrt{3(1-{x}^{2})}]dx$, $-1<x<1,y(0)=0$. If $y(\frac{1}{2})=\frac{m}{n},m$ and $n$ are coprime numbers, then $m+n$ is equal to __________.

Let ${x}$ denote the fractional part of $x$ and $f(x)=\frac{{\mathrm{cos}}^{-1}(1-{{x}}^{2}){\mathrm{sin}}^{-1}(1-{x})}{{x}-{{x}}^{3}},x\neq 0$. If $L\text{and}R$ respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x=0$, then $\frac{32}{{\pi }^{2}}({L}^{2}+{R}^{2})$ is equal to __________.

Let $\alpha|x|=|y| \mathrm{e}^{x y-\beta}, \alpha, \beta \in \mathbf{N}$ be the solution of the differential equation $x \mathrm{~d} y-y \mathrm{~d} x+x y(x \mathrm{~d} y+y \mathrm{~d} x)=0$, $y(1)=2$. Then $\alpha+\beta$ is equal to ________

Let the area of the region enclosed by the curves $y=3 x, 2 y=27-3 x$ and $y=3 x-x \sqrt{x}$ be $A$. Then $10 A$ is equal to

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{{4}^{x}}{{4}^{x}+2}$ and $M=\int _{f(a)}^{f(1-a)}x{\mathrm{sin}}^{4}(x(1-x))dx$, $N=\int _{f(a)}^{f(1-a)}{\mathrm{sin}}^{4}(x(1-x))dx;a\neq \frac{1}{2}$. If $\alpha M=\beta N,\alpha ,\beta \in \mathbb{N}$, then the least value of ${\alpha }^{2}+{\beta }^{2}$ is equal to ______

If $\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cossc}^2 x+\frac{3}{2}\right)+\beta \log _\epsilon\left|\tan \frac{x}{2}\right|+C$ where $\alpha, \beta \in \mathbb{R}$ and $\mathrm{C}$ is the constant of integration, then the value of $8(\alpha+\beta)$ equals _______

Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y-y={Y}^{'}(x)(X-x)$ and the co-ordinate axes, where $(x,y)$ is any point on the curve, is always $\frac{-{y}^{2}}{2{Y}^{'}(x)}+1,{Y}^{'}(x)\neq 0$. If $Y(1)=1$, then $12Y(2)$ equals ________.

Let $f:\rightarrow R\rightarrow (0,\infty )$ be strictly increasing function such that $\underset{x\rightarrow \infty }{\mathrm{lim}}\frac{f(7x)}{f(x)}=1$. Then, the value of $\underset{x\rightarrow \infty }{\mathrm{lim}}[\frac{f(5x)}{f(x)}-1]$ is equal to

If the function $f(x)=2 x^3-9 x^2+12 \mathrm{a}^2 x+1, \mathrm{a}>0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^2$, then $\alpha$ and $\alpha^2$ are the roots of the equation :

Let $\lim _{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right.$ $\left.+\ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right)$ be $\frac{\pi}{k}$, using only the principal values of the inverse trigonometric functions. Then $\mathrm{k}^2$ is equal to ________

If the value of the integral${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(\frac{{x}^{2}\mathrm{cos}x}{1+{\pi }^{x}}+\frac{1+{\mathrm{sin}}^{2}x}{1+{e}^{{(\mathrm{sin}x)}^{2023}}})dx=\frac{\pi }{4}(\pi +a)-2,$ then the value of $a$ is

The integral $\int \frac{({x}^{8}-{x}^{2})\mathrm{dx}}{({x}^{12}+3{x}^{6}+1){\mathrm{tan}}^{-1}({x}^{3}+\frac{1}{{x}^{3}})}$ is equal to :

If $\int \frac{{\mathrm{sin}}^{\frac{3}{2}}x+{\mathrm{cos}}^{\frac{3}{2}}x}{\sqrt{{\mathrm{sin}}^{3}x{\mathrm{cos}}^{3}x\mathrm{sin}(x-\theta )}}dx=A\sqrt{\mathrm{cos}\theta \mathrm{tan}x-\mathrm{sin}\theta }+B\sqrt{\mathrm{cos}\theta -\mathrm{sin}\theta \mathrm{cot}x}+C,$ where $C$ is the integration constant, then $AB$ is equal to

Let $g(x)$ be a linear function and $f(x)={\begin{matrix}g(x), & x\leq 0 \\ {(\frac{1+x}{2+x})}^{\frac{1}{x}}, & x>0\end{matrix}$, is continuous at $x=0$. If ${f}^{'}(1)=f(-1)$, then the value of $g(3)$ is

The area (in square units) of the region bounded by the parabola ${y}^{2}=4(x-2)$ and the line $y=2x-8$.

If the function $f(x)= \begin{cases}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x \neq 0 \\ a \log _e 2 \log _e 3 & , x=0\end{cases}$ is continuous at $x=0$, then the value of $a^2$ is equal to

Let $y=y(x)$ be the solution of the differential equation $\left(1+x^2\right) \frac{d y}{d x}+y=e^{\tan ^{-1} x}$, $y(1)=0$. Then $y(0)$ is

Let $y=y(x)$ be the solution curve of the differential equation $\sec y \frac{\mathrm{d} y}{\mathrm{~d} x}+2 x \sin y=x^3 \cos y, y(1)=0$. Then $y(\sqrt{3})$ is equal to :

Let $f,g:(0,\infty )\rightarrow R$ be two functions defined by $f(x)={\int }_{-x}^{x}(|t|-{t}^{2}){e}^{-{t}^{2}}dt$ and $g(x)={\int }_{0}^{{x}^{2}}{t}^{\frac{1}{2}}{e}^{-{t}^{2}}dt$. Then the value of $9(f(\sqrt{{\mathrm{log}}_{e}9}+g(\sqrt{{\mathrm{log}}_{e}9}))$ is equal to

The temperature $T(t)$ of a body at time $t=0$ is ${160}^{^{\circ}}F$ and it decreases continuously as per the differential equation $\frac{dT}{dt}=-K(T-80)$, where $K$ is positive constant. If $T(15)={120}^{^{\circ}}F$, then $T(45)$ is equal to

One of the points of intersection of the curves $y=1+3 x-2 x^2$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2}, 2\right)$. Let the area of the region enclosed by these curves be $\frac{1}{24}(l \sqrt{5}+\mathrm{m})-\mathrm{n} \log _{\mathrm{e}}(1+\sqrt{5})$, where $l, \mathrm{~m}, \mathrm{n} \in \mathbf{N}$. Then $l+\mathrm{m}+\mathrm{n}$ is equal to

The function $f(x)=2x+3{x}^{\frac{2}{3}},x\in R$, has

The area of the region enclosed by the parabola $y=4x-{x}^{2}$ and $3y={(x-4)}^{2}$ is equal to

Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 x}{\left(1+x^2\right)^2} y=x \mathrm{e}^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0 .$ Then the area enclosed by the curve $f(x)=y(x) \mathrm{e}^{-\frac{1}{\left(1+x^2\right)}}$ and the line $y-x=4$ is__________

The area of the region enclosed by the parabolas $y=x^2-5 x$ and $y=7 x-x^2$ is