Indefinite Integration

Let $\beta$ be the area of the region enclosed between the curves $C_1$, $C_2$, and the lines $x=\alpha_1$ and $x=\alpha_4$. Then the value of
$$-\frac1\pi\log_e\left(\beta-2\,e^{-\frac\pi2}\right)$$
is ________.

Let $f(x)=\displaystyle\int\left(\frac{16x+24}{x^2+2x-15}\right)dx$. If $f(4)=14\log_e(3)$ and $f(7)=\log_e\left(2^\alpha\cdot3^\beta\right)$, $\alpha,\beta\in\mathbb N$, then $\alpha+\beta$ is equal to :

The area of the region $\{(x,y):y\le\pi-|x|,\ y\le|x\sin x|,\ y\ge0\}$ is:

Let $y=y(x)$ be the solution of the differential equation $\dfrac{dy}{dx}=\left(1+x+x^2\right)\left(1-y+y^2\right)$, $y(0)=\dfrac12$. Then $(2y(1)-1)$ is equal to

The value of the integral $\displaystyle\int_{\pi/6}^{\pi/3}\left(\frac{4-\operatorname{cosec}^2x}{\cos^4x}\right)dx$ is:

Let $y=y(x)$ be the solution of the differential equation
$$(\tan x)^{1/2}\,dy=\left(\sec^3x-(\tan x)^{3/2}\,y\right)dx,\quad 0<x<\frac\pi2,\quad y\left(\frac\pi4\right)=\frac{6\sqrt2}5.$$
If $y\left(\dfrac\pi3\right)=\dfrac45\alpha$, then $\alpha^4$ equals ________.

Let $y=y(x)$ be the solution of the differential equation $\left(x^2-x\sqrt{x^2-1}\right)dy+\left(y\left(x-\sqrt{x^2-1}\right)-x\right)dx=0$, $x\ge1$. If $y(1)=1$, then the greatest integer less than $y\left(\sqrt5\right)$ is ________.

The value of the integral $\displaystyle\int_0^2\frac{\sqrt{x\left(x^2+x+1\right)}}{\left(\sqrt{x+1}\right)\left(\sqrt{x^4+x^2+1}\right)}\,dx$ is equal to :

Let $y=y(x)$ be the solution of the differential equation $x\sqrt{1-x^2}\,dy+\left(y\sqrt{1-x^2}-x\cos^{-1}x\right)dx=0$, $x\in(0,1)$, $\displaystyle\lim_{x\to1^-}y(x)=1$. Then $y\left(\dfrac12\right)$ equals :

If

$$\alpha = \int_{1/2}^{2}\frac{\tan^{-1}x}{2x^2 - 3x + 2}\,dx,$$

then the value of $\sqrt7\tan\left(\dfrac{2\alpha\sqrt7}{\pi}\right)$ is _____.

(Here, the inverse trigonometric function $\tan^{-1}x$ assumes values in $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$.)

Consider the equation
$$\int_1^e \frac{(\log_e x)^{1/2}}{x\left(a - (\log_e x)^{3/2}\right)^2}\,dx = 1, \qquad a \in (-\infty, 0)\cup(1,\infty).$$
Which of the following statements is/are TRUE?

If $y(x)$ is the solution of the differential equation
$$x\,dy - (y^2 - 4y)\,dx = 0 \quad\text{for } x > 0, \qquad y(1) = 2,$$
and the slope of the curve $y = y(x)$ is never zero, then the value of $10\,y(\sqrt2)$ is _____.

If $x=x(y)$ is the solution of the differential equation $y\dfrac{dx}{dy}=2x+y^3(y+1)e^y$, $x(1)=0$; then $x(e)$ is equal to :

$$\lim_{n\to\infty}\left(\frac{n^2}{\left(n^2+1\right)(n+1)}+\frac{n^2}{\left(n^2+4\right)(n+2)}+\frac{n^2}{\left(n^2+9\right)(n+3)}+\dots+\frac{n^2}{\left(n^2+n^2\right)(n+n)}\right)$$
is equal to

The value of $\displaystyle\int_0^\pi\frac{e^{\cos x}\sin x}{\left(1+\cos^2x\right)\left(e^{\cos x}+e^{-\cos x}\right)}dx$ is equal to :

Let $g:(0,\infty)\to\mathbb R$ be a differentiable function such that
$$\int\left(\frac{x(\cos x-\sin x)}{e^x+1}+\frac{g(x)\left(e^x+1-xe^x\right)}{\left(e^x+1\right)^2}\right)dx=\frac{x\,g(x)}{e^x+1}+c,$$
for all $x>0$, where $c$ is an arbitrary constant. Then :

The value of $b>3$ for which $12\displaystyle\int_3^b\frac1{\left(x^2-1\right)\left(x^2-4\right)}dx=\log_e\left(\frac{49}{40}\right)$, is equal to ______.

If $\displaystyle\int\frac1x\sqrt{\frac{1-x}{1+x}}\,dx=g(x)+c$, $g(1)=0$, then $g\left(\dfrac12\right)$ is equal to :

The integral $\dfrac{24}\pi\displaystyle\int_0^{\sqrt2}\frac{\left(2-x^2\right)dx}{\left(2+x^2\right)\sqrt{4+x^4}}$ is equal to ______.

If $\displaystyle\int\frac{\left(x^2+1\right)e^x}{(x+1)^2}dx=f(x)e^x+C$, where C is a constant, then $\dfrac{d^3f}{dx^3}$ at $x=1$ is equal to :