Calculus PYQ — Page 25

If $y=y(x)$ is the solution of the differential equation $x\frac{dy}{dx}+2y=x{e}^{x},y(1)=0$ then the local maximum value of the function $z(x)={x}^{2}y(x)-{e}^{x},x\in R$ is

The slope of normal at any point $(x,y),x>0,y>0$ on the curve $y=y(x)$ is given by $\frac{{x}^{2}}{xy-{x}^{2}{y}^{2}-1}$. If the curve passes through the point $(1,1)$, then $e\cdot y(e)$ is equal to

The integral $\int \frac{(1-\frac{1}{\sqrt{3}})(\mathrm{cos}x-\mathrm{sin}x)}{(1+\frac{2}{\sqrt{3}}\mathrm{sin}2x)}dx$ is equal to

Let $g:(0,\infty )\rightarrow R$ be a differentiable function such that $\int (\frac{x(\mathrm{cos}x-\mathrm{sin}x)}{{e}^{x}+1}+\frac{g(x)({e}^{x}+1-x{e}^{x})}{{({e}^{x}+1)}^{2}})dx=\frac{xg(x)}{{e}^{x}+1}+C$, for all $x>0$, where $C$ is an arbitrary constant. Then

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

$I={\int }_{\frac{\pi }{4}}^{\frac{\pi }{3}}(\frac{8\mathrm{sin}x-\mathrm{sin}2x}{x})dx$. Then

Let $f(x)=\mathrm{min}{[x-1],[x-2],\ldots ,[x-10]}$ where $[t]$ denotes the greatest integer $\leq t$. Then ${\int }_{0}^{10}f(x)dx+{\int }_{0}^{10}{(f(x))}^{2}dx+{\int }_{0}^{10}|f(x)|dx$ is equal _______. to

If ${\int }_{0}^{2}(\sqrt{2x}-\sqrt{2x-{x}^{2}})dx=$ ${\int }_{0}^{1}(1-\sqrt{1-{y}^{2}}-\frac{{y}^{2}}{2})dy+{\int }_{1}^{2}(2-\frac{{y}^{2}}{2})dy+I$, then $I$ equal to

The value of the integral $\frac{48}{{\pi }^{4}}{\int }_{0}^{\pi }(\frac{3\pi {x}^{2}}{2}-{x}^{3})\frac{\mathrm{sin}x}{1+{\mathrm{cos}}^{2}x}dx$ is equal to ______.

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

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

Let $f:[0,1]\rightarrow R$ be a twice differentiable function in $(0,1)$ such that $f(0)=3$ and $f(1)=5$. If the line $y=2x+3$ intersects the graph of $f$ at only two distinct points in $(0,1)$, then the least number of points $x\in (0,1)$, at which ${f}^{''}(x)=0$, is

The area of the bounded region enclosed by the curve $y=3-|x-\frac{1}{2}|-|x+1|$ and the $x$-axis is

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+(\frac{2{x}^{2}+11x+13}{{x}^{3}+6{x}^{2}+11x+6})y=\frac{(x+3)}{x+1},x>-1$, which passes through the point $(0,1)$. Then $y(1)$ is equal to

If $\frac{dy}{dx}+2y\mathrm{tan}x=\mathrm{sin}x,0<x<\frac{\pi }{2}$ and $y(\frac{\pi }{3})=0$, then the maximum value of $y(x)$ is

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{4{y}^{3}+2y{x}^{2}}{3x{y}^{2}+{x}^{3}},y(1)=1$. If for some $n\in N,y(2)\in [n-1,n)$, then $n$ is equal to _______.

Let the solution curve of the differential equation $x\frac{dy}{dx}-y=\sqrt{{y}^{2}+16{x}^{2}},y(1)=3$ be $y=y(x)$. Then $y(2)$ is equal to

If $y=y(x)$ is the solution of the differential equation $(1+{e}^{2x})\frac{dy}{dx}+2(1+{y}^{2}){e}^{x}=0$ and $y(0)=0$, then $6(y'(0)+{(y({\mathrm{log}}_{c}\sqrt{3}))}^{2})$ is equal to:

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+\frac{\sqrt{2}y}{2{\mathrm{cos}}^{4}x-\mathrm{cos}2x}=x{e}^{{\mathrm{tan}}^{-1}(\sqrt{2}\mathrm{cot}2x)},0<x<\frac{\pi }{2}$ with $y(\frac{\pi }{4})=\frac{{\pi }^{2}}{32}$. If $y(\frac{\pi }{3})=\frac{{\pi }^{2}}{18}{e}^{-{\mathrm{tan}}^{-1}(\alpha )}$, then the value of $3{\alpha }^{2}$ is equal to ______.

Let $y=y(x)$ be the solution of the differential equation $(1-{x}^{2})dy=(xy+({x}^{3}+2)\sqrt{1-{x}^{2}})dx,-1<x<1$ and $y(0)=0$. If ${\int }_{-\frac{1}{2}}^{\frac{1}{2}}\sqrt{1-{x}^{2}}y(x)dx=k$ then ${k}^{-1}$ is equal to

Let the solution curve $y=y(x)$ of the differential equation $(4+{x}^{2})dy-2x({x}^{2}+3y+4)dx=0$ pass through the origin. Then $y(2)$ is equal to _____.

Let $y=y(x)$ be the solution of the differential equation $(x+1){y}^{'}-y={e}^{3x}{(x+1)}^{2}$, with $y(0)=\frac{1}{3}$. Then, the point $x=-\frac{4}{3}$ for the curve $y=y(x)$ is

Let ${a}_{n}={\int }_{-1}^{n}(1+\frac{x}{2}+\frac{{x}^{2}}{3}+\ldots +\frac{{x}^{n-1}}{n})dx$ for every $n\in N$. Then the sum of all the elements of the set ${n\in N:{a}_{n}\in (2,30)}$ is _________.

Let $y=y(x)$ be the solution curve of the differential equation $\frac{dy}{dx}+\frac{1}{{x}^{2}-1}y={(\frac{x-1}{x+1})}^{\frac{1}{2}}$, $x>1$ passing through the point $(2,\sqrt{\frac{1}{3}})$. Then $\sqrt{7}y(8)$ is equal to