Integral Calculus Questions (265)

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

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

The value of the definite integral $\displaystyle\int_{-2}^{2}x^3\ln(1^x+3^x+5^x+15^x)\,dx$

Let $f$ be a function defined by $f(x)=\begin{cases}(x-r)^2, & r-1\leq x<r+1\\ 1, & r+1\leq x\leq r+2\end{cases}$ where $r=3k$, $k\in I$. Find $\displaystyle\sqrt{\int_0^{45}f(x)\,dx}$

If $\displaystyle\int\frac{\sec^2 x-2010}{\sin^{2010}x}\,dx=\frac{P(x)}{(\sin x)^{2010}}+C$, then the value of $P\!\left(\dfrac{\pi}{3}\right)$ is

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

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

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

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

If $f'(x) = \frac{1}{-x + \sqrt{x^2 + 1}}$ and $f(0) = -\frac{1 + \sqrt{2}}{2}$, then the value of $f(5)$ will be :

The integral $\int \frac{\sec^{3/2}\theta - \sec^{1/2}\theta}{2 + \tan^2\theta} \tan\theta d\theta$ is:

If $f(x) = \lim_{n \to \infty} \frac{x^n - x^{-n}}{x^n + x^{-n}}$, $0 < x < 1$, $n \in \mathbb{N}$ then $\int (\sin^{-1} x) f'(x) dx$ is equal to:

The derivative of $x^4 + x^{-5}$ is $-\left(4x^{-5} + 5x^{-6}\right)$. So, $$\int \frac{5x^3 + 4x^5}{\left(x^5 + x + 1\right)^2} dx =$$

If the anti-derivative of $\frac{x^3}{\sqrt{4+2x^2}}$ which passes through $(1, 2)$ is $\frac{1}{m}\left(1+2x^2\right)^{1/2}\left(x^2-1\right)+c$. Then:

Let $f(xy) = f(x) \cdot f(y)$, $\forall x > 0, y > 0$ and $f(1 + x) = 1 + x[1 + g(x)]$, where $\lim_{x \to 0} g(x) = 0$, then $$\int \frac{f(x)}{f'(x)} dx$$ is:

If $$\int \frac{dx}{x^2(x^n + 1)^{(n-1)/n}} = -[f(x)]^{1/n} + C$$, then $f(x)$ is