Calculus Questions (384)

Comment upon the nature of roots of the quadratic equation $x^2 + 2x + k = \int_0^k |1+k| dr$ depending on the value of $k \in \mathbb{R}$.

In which of the following cases the given equations has atleast one root in the indicated interval?

Let $f: \mathbb{R} \to \mathbb{R}$ be a function satisfying $f(x+2y) = f(x)e^{2y} + f(2y)e^x + x^2\left(1-e^{2y}\right) + 4y^2\left(1-e^x\right) + 4xyy$ for all $x, y \in \mathbb{R}$ and $f'(0) = 1$, then:

Let $I = \int_{\pi/4}^{\pi/3} \frac{\sin x}{x} dx$, then $I$ belongs to:

If $I_{m,n} = \int \cos^m x \sin nx dx$, then $7I_{4,3} - 4I_{3,2} =$$

$\lim f(x)$ does not exist when (where $[x]$ denotes the greatest integer less than or equal to $x$)

A function $f(x)$ continuous on $\mathbb{R}$ and periodic with $2\pi$ satisfies $f(x) + (\sin x) f(x + \pi) = \sin^2 x$ then,

Assume that $\lim_{\theta \to 1} f(0)$ exists and $\frac{\theta^2 + 0 - 2}{\theta + 3} \leq \frac{f(0)}{\theta^2} \leq \frac{\theta^2 + 20 - 1}{\theta + 3}$ holds for certain interval containing the point $\theta = -1$ then $\lim_{\theta \to 1} f(0)$ and $\lim_{\theta \to 1} \frac{f(0)}{\theta^2}$ is :

Let $f: R \to R$ be defined by $f(x) = \begin{cases} x + 2x^2 \sin \frac{1}{x} & \text{for } x \neq 0 \\ 0 & \text{for } x = 0 \end{cases}$ then

The value of $\int_1^8 x\sin[x^2 - \pi] dx$, where $[.]$ denotes the greatest integer function is:

If $\int \frac{dx}{x^2\left(x^7 - 6\right)} = A\left[\ln\left(p^9 + 9p^2 - 2p^3 - 18p\right)\right] + c$, then:

Let a function $f$ be defined as $f(x) = \begin{cases} \frac{|x-1|}{x^2+1} & \text{if } x > -1 \\ x^2 & \text{if } x \leq -1 \end{cases}$. Then the number of critical point(s) on the graph of this function is/are:

If $I = \int \frac{\sin x + \sin^3 x}{\cos 2x} dx = P\cos x + Q\ln |f(x)| + R$, then:

If $\int \sqrt{\cos ecx + 1}\,dx = kfog(x) + c$, where $k$ is a real constant, then :

A function is defined as $f(x) = [\tan x] + \sqrt{\tan x - [\tan x]}$ $0 \leq x 2 \\ 5x - 7 & \text{if } x \leq 2 \end{cases}$ then:

Anti-derivative of $$\frac{x - 1}{(x + 1)\sqrt{x^3 + x^2 + x}}$$ is:

If $F(x) = f(x)g(x)$ and $f'(x)g'(x) = c$, then (where $f$ and $g$ are thrice differentiable)

Area bounded by the curves $y = \left[\frac{x^2}{64} + 2\right]$ ([$.$] denotes the greatest integer function), $y = x - 1$ and $x = 0$ above the $x$-axis is:

If $f(x) = x + \int_0^x (y^2 + x^2)f(y) dy$, then:

If $m, n$ are even integers and $p, q \in \mathbb{R}$, then $\int_{p+ma}^{q+na} g(t)dt$ is equal to:

Area bounded by $x^2-y-1=0$, $y=-1$ and $x=0$ (positive side) is

If $\Lim_{n \to \infty} \frac{1}{n^2} \sum_{k=1}^{n-1} k \left[ \int_0^{k/n} \sqrt{(x-k)(k+1-x)} dx \right] = \frac{\pi}{m^n}$, then:

If $A = \int_0^{\sin \theta} \frac{t dt}{1 + t^2}$ and $B = \int_0^{\cos \theta} \frac{dt}{t(1 + t^2)}$, then the value of $e^A e^B \begin{vmatrix} A & A^2 & B \\ 1 & B^2 & -1 \\ 1 & A^2 + B^2 & -1 \end{vmatrix}$ is:

Let $f$ be a function defined on $(-\pi/2, \pi/2)$ as follows: $f(x) = \begin{cases} \frac{2^{[1/n]} - [x] - \frac{[x]}{[n2-1]}}{x\tan x} & x \neq 0 \\ k & x = 0 \end{cases}$. The value of $k$ so that $f$ is continuous at $x = 0$ is:

$I = \int \frac{dx}{(\sin x - 2\cos x)(2\cos x + \sin x)}$ is equal to

Let $f(x)$ be a continuous function and 'c' is a constant satisfying $\int_0^x f(t) dt = e^x - ce^{2x} \int_0^x f(t)^2 dt$, then:

Let $f(x) = \lim_{n \to \infty} \frac{2x^{2n} \sin + x}{1 + x^{2n}}$ then which of the following alternative(s) is/are correct?

If $f(x) = \begin{cases} \frac{x \cdot \ln(\cos x)}{\ln(1+x^2)} & x \neq 0 \\ 0 & x = 0 \end{cases}$ then:

$$\int \frac{dx}{\prod_{i=0}^{n}(x+r)} \text{ is equal to:}$$

If $f(x)$ is an even function, then:

$\text{Lim}_{x \to 0^+} \left[3f\left(\frac{x^3-\sin^3 x}{x^4}\right)-f\left(\left[\frac{\sin x^3}{x}\right]\right)\right]$ where $[\cdot]$ denote greatest integer function.

Area enclosed by $y=g(x)$, $x=1$ and $x=37$, where $g(x)$ is the inverse of $f(x)=x^3+3x+1$, is

Let $f(x)$ is a polynomial function and $(f(x))^2 + (f'(x))^2 = 0$, then find $\lim_{x \to 0}\frac{f(x)}{f'(x)}\left[\frac{f'(x)}{f(x)}\right]$, (where [.] denotes greatest integer function) is_____.

A lane of width $27m$ runs at right angle out of a road of $64m$. The maximum length of a pole which can be carried from the road to the lane keeping it horizontal is $L$, then $\sqrt[3]{L}$ equals to _____.

Let $f(x)$ is a quadratic function such that $f(0) = 1 \& f(-1) = 4$. If $\int \frac{f(x)dx}{x^2(x + 1)^2}$ is a rational function, then $f(10) = $

If $f(x)$ is even and periodic with period $T$, $\int_0^a f(x)dx=3$ and $\int_{-T/2}^{3T/2}f(x)dx=18$, then $\int_{-a}^{a+5T}f(x)dx$ is

If $f(x) = \sin x + \displaystyle\int_{-\pi/2}^{\pi/2}(\sin x + t\cos x)f(t)\,dt$, then $f(x)$ may be equal to $\left(-\dfrac{1}{k}\sin x - \dfrac{2}{k}\cos x\right)$, where $k$ is a numerical quantity which equals

The area of the region bounded by $y=x^2$ and $y=\sec^{-1}[-\sin^2 x]$, where $[\cdot]$ is the GIF, is

Value of $\displaystyle\int_0^1 \frac{\sin x}{x}\,dx$ lies in the interval

If $f(x) = \begin{vmatrix}\cos x & e^{x^2} & 2x\cos^2(x/2)\\ x^2 & \sec x & \sin x+x^3\\ 1 & 2 & x+\tan x\end{vmatrix}$ and $\displaystyle\int_{-\pi/2}^{\pi/2}(1+x^4)(f(x)+f''(x))\,dx = 2\lambda+3$, then $\lambda$ is

The value of $\int_0^1 \lim_{n \to \infty} \sum_{k=0}^n \frac{x^{k+2^k}}{k!} dx$ is:

If $a$ is a positive integer, then the number of values of $a$ satisfying $$\int_0^{\pi/2} \left[a^2\left(\frac{\cos 3x}{3} + \cos x\right) + a\sin x - 20\cos x\right] dx \leq -\frac{a^2}{3}$$ is:

Let $f(x)=\begin{cases}|1-2x^2|, & 0\le x<1 \\ [x^2-2x], & 1\le x<2\end{cases}$. If $m,n$ are number of points of discontinuity and non-differentiability of $f(x)$ in $(0,2)$, then

Water is filled at rate $\pi$ cm$^3$/s in right circular conical vessel (vertex up) of height 5 cm and diameter 8 cm. When water height is 3 cm, rate of increase of wet conical surface area is (cm$^2$/s)

$$\int \frac{x^4 + 1}{x^4 + 1} dx =$$

If $\int \left[\left(\frac{x}{e}\right)^x + \left(\frac{e}{x}\right)^x\right] \ln udx = A\left(\frac{x}{e}\right)^x + B\left(\frac{e}{x}\right)^x + C$, then the value of $A + B$ is

The equation $1012x^{2023}-12138x^{2022}-119x+714=0$ has a root in $(a^{1/2022},b^{1/3})$; $a,b\in\mathbb{N}\geq2$. The value of $4\displaystyle\int_{\sqrt{a}}^{b^{1/3}}\frac{x\cos x^2}{\cos x^2+\cos(263-x^2)}\,dx$ is

If $\int \frac{dx}{\sqrt{x + 7} - \sqrt[3]{x + 7}} = P\sqrt{x + 7} + Q\sqrt[3]{x + 7} + R\ln|x + 7|^{1/4}| + c$. Then find the value of $P + Q + R$.

Let $f$ and $g$ be continuously differentiable functions such that $f(0) = 0, f'(0) = 2$ and $g(x) = f(-x + f(x))). The value of $g'(0)$ equals.

The function $f:[0,1] \to [0,1]$ is continuous and has the property $f(f(x)) = 1-x$ for all $x \in [0,1]$ and $\alpha = \int_0^1 f(x)dx$, then: