Integral Calculus Questions (265)

The intercepts on the x-axis made by tangents to the curve $y=\displaystyle\int_0^x|t|\,dt$, $x\in\mathbb{R}$, which are parallel to $y=2x$, are equal to

Let $f(x) = x + \sin x$. Suppose $g$ denotes the inverse function of $f$. If the value of $g'\left(\frac{\pi}{4} + \frac{1}{\sqrt{2}}\right)$ is $l$ then $2l = $

Let $e$ be the eccentricity of a hyperbola and $f(e)$ be the eccentricity of its conjugate hyperbola. Then $\int(f(e)+f(f(e)))de=g(e)$ and $g(\sqrt{2})=2$, then $g(e)=$

Let $I_1=\int_0^x e^{tx}\cdot e^{-t^2}\,dt$ and $I_2=\int_0^x e^{-t^2/4}\,dt$ where $x>0$. Then the value of $\dfrac{I_1}{I_2}$ is

$\displaystyle\int_1^{81}\dfrac{dx}{\sqrt{x}+\sqrt[4]{x}}$

Consider: I) If $f(x)$ is bounded for $x\in[a,b]$, then area bounded by $y=f(x)$, $x$-axis, $x=a$ and $x=b$ is $\int_a^b f(x)dx$. II) If $f(x)$ is bounded and differentiable on $[a,b]$, then the area between $y=f(x)$ and $y=f^{-1}(x)$ is equal to double the value of $\int_a^b|f(x)-x|dx$.

If $x\displaystyle\int_0^x f(t)\,dt = (x+1)\int_0^x tf(t)\,dt$ for $x>0$, and $f(1)=\dfrac{1}{e}$, then $f(-1)$ is

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

$\int_{-5}^{5}\dfrac{x^{10}}{x^{10}+(5-x)^{10}}\cdot\dfrac{dx}{x^2+16}$

Let $y=f(x)$ be a differentiable function satisfying $\int_2^x f(t)\,dt+2=\dfrac{x^2}{2}+\int_x^2 t^2 f(t)\,dt$. Then $\int_{-\pi/4}^{\pi/4}\dfrac{f(x)+x^9-x^3+x+1}{\cos^2 x}\,dx=$

$\displaystyle\int_0^{10}[x^2]dx$ (where $[.]$ is GIF) equals

$\displaystyle\int\sqrt{\dfrac{x}{1-x^3}}dx=$

The area enclosed between the curves $y=ax^2$ and $x=ay^2$ ($a>0$) is 1 sq. unit. Then the value of $a$ is

$\lim_{n\to\infty}\left(\dfrac{n^2}{n^2+1}\cdot\dfrac{n^2}{n^2+4}\cdots\dfrac{n^2}{n^2+n^2}\right)^{1/n}$

The area bounded by the curve $y = \dfrac{1}{x^2 - 2x + 2}$ and the $x$-axis equals

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

$\displaystyle\int_0^1\ln\left(\dfrac{1}{x}-1\right)dx$

The area of the region between $y=\sqrt{\dfrac{1+\sin x}{\cos x}}$ and $y=\sqrt{\dfrac{1-\sin x}{\cos x}}$ bounded by $x=0$ and $x=\pi/4$ is

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

The area bounded by $y=x^2-3$ and $y=kx+2$ is the least. Then $k$ is

$\displaystyle I=\int_{-\pi/2}^{\pi/2}\dfrac{8\sqrt{2}\sin x}{(1+e^x)(1+\sin^4 x)}dx$

Let $S$ be the region bounded by the curves $y=x^3$ and $y^2=x$. The curve $y=2|x|$ divides $S$ into two regions of areas $R_1$ and $R_2$. If $\max\{R_1,R_2\}=R_2$, then $\dfrac{R_2}{R_1}=$

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

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

If $\int\dfrac{5\tan x\,dx}{\tan x-2}=x+a\ln|\sin x-2\cos x|+C$, then $a$ is equal to ($C$ is constant of integration)

The area bounded by $y=f(x)=x^4-2x^3+x^2+3$, the $x$-axis, and the lines $x=0$ to $x=3$ is $A$. Find $10A/3$.

Let $[t]$ denote greatest integer $\le t$. If $f(x)=\int_0^x\left(\left[\frac{1}{1-t^2}\right]+\left[\frac{1}{t^2-1}\right]\right)dt$, then $f\left(\frac{\sqrt{5}}{2}\right)=$

Let $P(x)$ be a quadratic polynomial with $P(1)=-1$. If $\displaystyle\int\frac{P(x)\,dx}{(2x-3)^2(3x-2)^2}=\frac{1}{5}\ln|f(x)|+C$ where $f(x)$ is rational and $\lim_{x\to\infty}f(x)=\frac{4}{3}$, then $|f(1)|$ is

Let $I(x)=\displaystyle\int\frac{x+1}{x(x^2e^{2x}-1)}\,dx=\frac{1}{4}\ln\frac{(xe^x)^\alpha-2(xe^x)^\beta+\gamma}{x^4e^{4x}}+c$, then $\alpha+\beta+\gamma$ equals

Let $f$ be continuous and differentiable in $(x_1, x_2)$. If $f(x)f'(x) \geq x\sqrt{1-[f(x)]^4}$ and $\lim_{x\to x_1}(f(x))^2=1$, $\lim_{x\to x_2}(f(x))^2=\frac{1}{2}$. Then minimum value of $\left[x_1^2 - x_2^2\right]$ is ........... (where $[\cdot]$ denotes GIF)

$\displaystyle\int_0^{\pi/2}\dfrac{\sin x-\cos x}{1+\sin x\cos x}dx=$

A strictly increasing continuous function $f(x)$ intersects its inverse $f^{-1}(x)$ at $x=\alpha$ and $x=\beta$, $\displaystyle\int_\alpha^\beta (f(x)+f^{-1}(x))\,dx=13$, where $\alpha,\beta\in\mathbb{N}$. Then $|\alpha\beta|$ equals

$\displaystyle\int\frac{6x^{10}+4}{x^3\sqrt{x^{10}-3x^4-1}}\,dx,\;x>0$

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 integral $\displaystyle\int_0^1\dfrac{\tan^{-1}x}{1+x}\,dx$ equals

Let $I=\displaystyle\int_0^2\!\left[\left|x^2-5x+4\right|+\left[\sin\frac{3\pi}{2}x\right]\right]dx$ (where $[\cdot]$ is GIF). Then $I+\dfrac{2}{3}$ is

Which must be true (in order) for: I) $\int e^{ax}\sin bx\,dx=\frac{e^{ax}}{a^2+b^2}(a\sin bx-b\cos bx)+c$, $a,b\in\mathbb{R}-\{0\}$. II) $\int\frac{f'(x)}{(f(x))^n}dx=\frac{(f(x))^{-n+1}}{-n+1}+c$, $n\ne 1$, $f(x)>0$. III) $\int e^{kx}\frac{f(kx)+f'(kx)}{k}dx=e^{kx}f(kx)+c$. IV) Let $F(x)$ be an indefinite integral of $\sin^2 x$, then $F(x+\pi)=F(x)$ for all real $x$.

For positive integer $n$, let $I_n=\displaystyle\int_{-\pi}^{\pi}\!\left(\frac{\pi}{2}-|x|\right)\cos nx\,dx$. Find $[I_1+I_2+I_3+I_4]$ (GIF).

If $f(x) = \lim_{n \to \infty} \frac{\tan(1/n)\log(1/n)}{n}$, and $\int \frac{f(x)}{\sqrt{\sin^{11} x \cos x}} dx = g(x) + C$ (C being the constant of integration). Then:

MATCH THE FOLLOWING: (A) $\int \frac{e^{2x} - 1}{e^{2x} + 1} dx$ is equal to (B) $\int \frac{1}{(e^x + e^{-x})^2} dx$ is equal to (C) $\int \frac{e^x}{1 + e^x} dx$ is equal to (D) $\int \frac{1}{\sqrt{1 - e^{2x}}} dx$ is equal to

A strictly increasing continuous function $f(x)$ intersects its inverse $f^{-1}(x)$ at $x=\alpha$ and $x=\beta$, $\displaystyle\int_\alpha^\beta (f(x)+f^{-1}(x))\,dx=13$, where $\alpha,\beta\in\mathbb{N}$. Then $|\alpha\beta|$ equals

If $I=\displaystyle\int\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}\,dx$, then $I$ equals

$\displaystyle\int\dfrac{2x^3+3x^2-11x-17}{(x^2-x-6)^2}dx$

If $f(x)=\displaystyle\int x^{1/3}(x^{2/3}+x^{1/3}+1)(2x^{2/3}+3x^{1/3}+6)^3\,dx$, $f(0)=0$, then $\dfrac{16}{11}f(1)$ is equal to

The intercepts on the x-axis made by tangents to the curve $y=\displaystyle\int_0^x|t|\,dt$, $x\in\mathbb{R}$, which are parallel to $y=2x$, are equal to

$\displaystyle\int_{-1}^{1}\dfrac{d}{dx}\left(\tan^{-1}\dfrac{1}{x}\right)dx=$

The smaller area (in sq. units) included between the curves $\sqrt{x}+\sqrt{|y|}=1$ and $|x|+|y|=1$ is

If $I=\displaystyle\int\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}\,dx$, then $I$ equals

If $\displaystyle\int (x^2+1)(x+1)^2 e^x\,dx = A(f(x))^2 + C$ and $f(-1) = \frac{2}{e}$, then $2A + f(0)$ equals

The area enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over the interval $\left[0,\dfrac{\pi}{2}\right]$ is