Calculus Questions (384)

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:

The function $f: (a, \infty) \to R$ where $R$ denotes the range corresponding to the given domain, with rule $f(x) = 2x^3 - 3x^2 + 6$ will have an inverse provided:

The least value of $'u'$ for which the equation, $\frac{4}{\sin x} + \frac{1}{1 - \sin x} = u$ has atleast one solution on the interval $(0, \pi/2)$ is:

The value of $a$ for which the equation $\int_0^a \sin^2\left(\frac{t}{2}\right)dt = a^2x^2 - \frac{1}{2}(3x-1) + \frac{1}{a^2}$ possesses a solution are:

If $f(x) = \frac{e^{x^2}(1+4x)^{1/2}}{\ln(1-x^2)}$ for $x \neq 0$, then $f$ has:

If $\int (x^9 + x^6 + x^3)(2x^6 + 3x^3 + 6)^{1/3} dx = a(2x^9 + 3x^6 + 6x^3)^{4/3} + c$, then the value of $48a$ must be

Let $P(x)$ be a polynomial of least degree whose graph has three points of inflection $(-1,-1), (1,1)$ and a point with abscissa $0$ at which the curve is inclined to the axis of abscissa at an angle of $60°$. Then $\int_0^1 P(x)dx$ equals to:

If $\int \frac{\cos^2 x + \sin 2x}{(2\cos x - \sin x)^2} dx = \frac{\cos x}{2\cos x - \sin x} + ax + b\ln|2\cos x - \sin x| + c$, then:

If $f(x) = 4x^3 - x^2 - 2x + 1$ and $g(x) = \begin{cases} \min\{f(t): 0 \leq t \leq x\}, & 0 \leq x \leq 1 \\ 3 - x & , 1 h''(x) h(x)$ for each $x \in J$. Then:

If $\int \frac{dx}{\sqrt{9x^2 + 4x + 6}}$ to evaluate $I$, one of the most proper substitution could be:

Suppose $f(x)$ is a function satisfying the following conditions:\n(i) $f(0) = 2, f(1) = 1$,\n(ii) $f$ has a minimum value at $x = 5/2$\n(iii) For all $x, f'(x) = \begin{vmatrix} 2ax & 2ax - 1 & 2ax + b + 1 \\ b & b + 1 & -1 \\ 2(ax + b) & 2ax + 2b + 1 & 2ax + b \end{vmatrix}$\nRange of $f(x)$ is

Let $f: \left[0, \frac{\pi}{2}\right] \to \mathbb{R}$ be such that $f(0) = 3$ and $f'(x) = \frac{1}{1 + \cos x}$. If $a < f\left(\frac{\pi}{2}\right) < b$, then $a$ and $b$ can be

If $I_n = \int \frac{dx}{(x^2 + a^2)^n}$, where $n \in \mathbb{N}$ and $n > 1$. If in and $I_{n-1}$ are related by the relation $PI_n = \frac{x}{(x^2 + a^2)^{n-1}} + QI_{n-1}$. Then $P$ and $Q$ are respectively given by:

The value of the integral $\int_0^\pi \frac{\sin(n+1/2)x}{\sin x/2} dx$ $(n \in \mathbb{N})$ is:

If $I$ is the greatest of the definite integrals $I_1 = \int_0^1 e^{-x}\cos^2 x dx, I_2 = \int_0^1 e^{-x^2}\cos^2 x dx, I_3 = \int_0^1 e^{-x^2} dx, I_4 = \int_0^1 e^{-x^2/2} dx$ then:

For $x>0$, $y>0$ with $x^2y^3=6$, find $\min(3x+4y)$.

Given $f(x) = \frac{e^x - \cos 2x - x}{x^2}$ for $x \in \mathbb{R} - \{0\}$, $\{x\}$ is fractional part function $$g(x) = \begin{cases} f\{x\} & n 1)$ is equal to :