Calculus PYQ — Page 2

$\int_{-1}^1 \frac{x^3 + |x| + 1}{x^2 + 2|x| + 1}dx$ is equal to

If the interval in which f(x) = $\frac{x}{4}$ + $\frac{4}{x}$, x ≠ 0 is strictly increasing is (-∞, a) ∪ (b, ∞), then

The marginal cost (MC) and marginal revenue (MR) functions of a product are $MC = 20 + \frac{x}{20}$ and $MR = 30$ respectively. If the fixed cost is 200, then the maximum value of the profit is:

$\int \frac{e^{2x} - 1}{e^{2x} + 1} dx =$

Match List-I with List-II (where $c$ is an arbitrary constant) | List-I | List-II | | --- | --- | | (A) $\int \tan x \, dx$ | (I) $\log\vert \sec x + \tan x\vert + c$ | | (B) $\int \cot x \, dx$ | (II) $\log\vert \sec x\vert + c$ | | (C) $\int \sec x \, dx$ | (III) $\log\vert \sin x\vert + c$ | | (D) $\int \cosec x \, dx$ | (IV) $\log\vert \cosec x - \cot x\vert + c$ | Choose the correct answer from the options given below:

The product of order and degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^2 + x^2y\left(\frac{d^2y}{dx^2}\right)^3 = 2x^5$ is:

The integral I = $\int e^x\left(\frac{x - 1}{3x^2}\right) dx$ is equal to

The value of the integral $I = \int_0^1 \frac{1}{\sqrt{x^2 + 2x + 3}} dx$ is:

The minimum value of the function $f(x) = x^3 + (10-x)^3$ occurs at:

The value of the definite integral $\int_0^1 e^x \frac{(1-x)^2}{(1+x^2)^2}dx$ is:

If $\int \frac{2x - 5}{(2x - 3)^3} e^{2x} dx = \frac{\lambda e^{2x}}{(2x - 3)^2} + C$, where $C$ is an arbitrary constant then the value of $\lambda$ is

The area bounded by the curve $y = \log x, y = 0$ and $x = e$, is

The area (in sq. units) of the region bounded by $y = -1, y = 2, x = y^3$ and $x = 0$ is equal to

The value of $\int_0^1 [\log x - \log(1-x)] dx$ is

$\int \frac{dx}{\sqrt{5 - 4x - x^2}}$ is equal to

The solution of the differential equation $\frac{dy}{dx} - \frac{y}{x} = 2\log_e x$

Differentiation of $\frac{x^3}{1 - x^3}$ with respect to $x^3$ is equal to:

$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals

The general solution of the differential equation $x\left(\frac{dy}{dx}\right) = y + x \tan\left(\frac{y}{x}\right)$ is

Match List-I with List-II | List-I | List-II | | --- | --- | | Definite integral | Value | | --- | --- | | (A) $\int_{1}^{e} \frac{\log x}{x} dx$ | (I) 4 | | (B) $\int_{-2}^{2} x^3(1 - x^2) dx$ | (II) $\frac{1}{2}$ | | (C) $\int_{1}^{2} x \, dx$ | (III) 0 | | (D) $\int_{-2}^{2} \vert x\vert dx$ | (IV) $\frac{3}{2}$ | Choose the correct answer from the options given below:

For $x > y > 0$, if $x^5 y^6 = (x + y)^{11}$, then $\frac{d^2y}{dx^2}$ is

The function $f(x) = x^4 - 2x^2$ is increasing on

The solution of the differential equation $\frac{dy}{dx} = \sqrt\frac{{y}}{x}$ is

A balloon which always remains spherical, has a variable diameter $\frac{3}{2}(5x+7)$. Then the rate of change of its volume with respect to x is