Calculus PYQ — Page 31

The value of ${\int }_{-\pi /2}^{\pi /2}\frac{{\mathrm{cos}}^{2}x}{1+{3}^{x}}dx$ is:

The value of the integral ${\int }_{-1}^{1}{\mathrm{log}}_{e}(\sqrt{1-x}+\sqrt{1+x})dx$ is equal to:

If the curve $y=y(x)$ is the solution of the differential equation $2({x}^{2}+{x}^{5/4})dy-y(x+{x}^{1/4})dx=2{x}^{9/4}dx,x>0$ which passes through the point $(1,1-\frac{4}{3}{\mathrm{log}}_{e}2),$ then the value of $y(16)$ is equal to

For $x>0$, if $f(x)={\int }_{1}^{x}\frac{{\mathrm{log}}_{e}t}{(1+t)}dt$, then $f(e)+f(\frac{1}{e})$ is equal to

Let ${I}_{n}={\int }_{1}^{e}{x}^{19}(\mathrm{log}|x|{)}^{n}dx,$ where $n\in N$. If $(20){I}_{10}=\alpha {I}_{9}+\beta {I}_{8},$ for natural numbers $\alpha$ and $\beta$, then $\alpha -\beta$ equal to _______.

Let $P(x)={x}^{2}+bx+c$ 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 divided by $(x-2)$ Then the value of $9(b+c)$ is equal to:

Let $F:[3,5]\rightarrow R$ be a twice differentiable function on $(3,5)$ such that $F(x)={e}^{-x}{\int }_{3}^{x}(3{t}^{2}+2t+4{F}^{'}(t))dt.$ If ${F}^{'}(4)=\frac{\alpha {e}^{\beta }-224}{{({e}^{\beta }-4)}^{2}}$, then $\alpha +\beta$ is equal to _____.

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}\frac{{x}^{3}}{(1-\mathrm{cos}2x{)}^{2}}{\mathrm{log}}_{e}(\frac{1+2x{e}^{-2x}}{{(1-x{e}^{-x})}^{2}}) & ,x\neq 0 \\ \alpha & ,x=0\end{matrix}$ If $f$ is continuous at $x=0,$ then $\alpha$ is equal to:

The area (in sq. units) of the region, given by the set ${(x,y)\in R\times R\mid x\geq 0,2{x}^{2}\leq y\leq 4-2x}$ is :

If the function $f(x)={\begin{matrix}\frac{1}{x}{\mathrm{log}}_{e}(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}),x<0 \\ k,x=0 \\ \frac{{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x-1}{\sqrt{{x}^{2}+1}-1},x>0\end{matrix}$ is continuous at $x=0,$ then $\frac{1}{a}+\frac{1}{b}+\frac{4}{k}$ is equal to :

$\underset{x\rightarrow 2}{\mathrm{lim}}(\sum _{n=1}^{9}\frac{x}{n(n+1){x}^{2}+2(2n+1)x+4})$ is equal to :

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\mathrm{sin}}^{-1}x-{\mathrm{tan}}^{-1}x}{3{x}^{3}}$ is equal to $L,$ then the value of $(6L+1)$ is

Let a function $g:[0,4]\rightarrow R$ be defined as $g(x)={\begin{matrix}\underset{0\leq t\leq x}{\mathrm{max}{{t}^{3}-6{t}^{2}+9t-3}}, & 0\leq x\leq 3 \\ 4-x, & 3<x\leq 4\end{matrix}$ then the number of points in the interval $(0,4)$ where $g(x)$ is NOT differentiable, is _________.

Let $f:R\rightarrow R$ be a function defined as $f(x)={\begin{matrix}\frac{\mathrm{sin}(a+1)x+\mathrm{sin}2x}{2x} & ,\mathrm{if}x<0 \\ b & ,\mathrm{if}x=0 \\ \frac{\sqrt{x+b{x}^{3}}-\sqrt{x}}{b{x}^{5/2}} & ,\mathrm{if}x>0\end{matrix}$ If $f$ is continuous at $x=0$, then the value of $a+b$ is equal to :

Let a function $f:R\rightarrow R$ be defined as, $f(x)={\begin{matrix}\mathrm{sin}x-{e}^{x} & \mathrm{if}x\leq 0 \\ a+[-x] & \mathrm{if}0<x<1 \\ 2x-b & \mathrm{if}x\geq 1\end{matrix}$ Where $[x]$ is the greatest integer less than or equal to $x.$ If $f$ is continuous on $R$, then $(a+b)$ is equal to:

If $\alpha ,\beta$ are the distinct roots of ${x}^{2}+bx+c=0,$ then $\underset{x\rightarrow \beta }{\mathrm{lim}}\frac{{e}^{2({x}^{2}+bx+c)}-1-2({x}^{2}+bx+c)}{(x-\beta {)}^{2}}$ is equal to

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}2\mathrm{sin}(-\frac{\pi x}{2}), & \mathrm{if}x<-1 \\ |a{x}^{2}+x+b|, & \mathrm{if}-1\leq x\leq 1 \\ \mathrm{sin}(\pi x), & \mathrm{if}x>1\end{matrix}$ If $f(x)$ is continuous on $R$, then $a+b$ equals :

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\mathrm{sin}}^{2}(\pi {\mathrm{cos}}^{4}x)}{{x}^{4}}$ is equal to :

If the integral ${\int }_{0}^{10}\frac{[\mathrm{sin}2\pi x]}{{e}^{x-[x]}}dx=\alpha {e}^{-1}+\beta {e}^{-\frac{1}{2}}+\gamma ,$ where $\alpha ,\beta ,\gamma$ are integers and $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of $\alpha +\beta +\gamma$ is equal to:

The range of $a\in R$ for which the function $f(x)=(4a-3)(x+{\mathrm{log}}_{e}5)+2(a-7)\mathrm{cot}(\frac{x}{2}){\mathrm{sin}}^{2}(\frac{x}{2}),x\neq 2n\pi ,n\in N$, has critical points, is :

A wire of length $36m$ is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $(\frac{4}{\pi }+1)k$ is equal to

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\int }_{0}^{{x}^{2}}(\mathrm{sin}\sqrt{t})dt}{{x}^{3}}$ is equal to:

The value of ${\int }_{-1}^{1}{x}^{2}{e}^{[{x}^{3}]}dx,$ where $[t]$ denotes the greatest integer $\leq t,$ is :

If $\int \frac{dx}{{({x}^{2}+x+1)}^{2}}=a{\mathrm{tan}}^{-1}(\frac{2x+1}{\sqrt{3}})+b(\frac{2x+1}{{x}^{2}+x+1})+C,x>0$ where $C$ is the constant of integration, then the value of $9(\sqrt{3}a+b)$ is equal to _________.