Calculus PYQ — Page 10

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 _____.

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

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

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

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

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

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

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:

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 :

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

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:

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:

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

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

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

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

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

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

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

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