Integration by parts

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 ________.

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

If $x=x(y)$ 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 :

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 :

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 :

If $n(2n+1)\displaystyle\int_0^1\left(1-x^n\right)^{2n}dx=1177\int_0^1\left(1-x^n\right)^{2n+1}dx$, then $n\in\mathbf N$ is equal to ______.

Let $I_n(x)=\displaystyle\int_0^x\frac1{\left(t^2+5\right)^n}dt$, $n=1,2,3,\ldots$. Then :

The value of $\dfrac{48S_2}{\pi^2}$ is ____.

Which of the following inequalities is/are TRUE?

Let $f:\mathbb R\to\mathbb R$ be a differentiable function such that its derivative $f'$ is continuous and $f(\pi) = -6$. If $F:[0,\pi]\to\mathbb R$ is defined by $F(x) = \int_0^x f(t)\,dt$, and if

$$\int_0^\pi\left(f'(x) + F(x)\right)\cos x\,dx = 2,$$

then the value of $f(0)$ is ______

The value of $\displaystyle\int_{-\pi/2}^{\pi/2}\frac{x^2\cos x}{1+e^x}\,dx$ is equal to

If $\displaystyle\int_1^3x^2F'(x)\,dx=-12$ and $\displaystyle\int_1^3x^3F''(x)\,dx=40$, then the correct expression(s) is(are)

The value of
$$\int_0^14x^3\left\{\frac{d^2}{dx^2}(1-x^2)^5\right\}dx$$
is

The integral $\displaystyle\int\left(1+x-\frac1x\right)e^{x+\frac1x}\,dx$ is equal to :

The integral $\displaystyle\int x\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)dx$ $(x>0)$ is equal to :

If for $n\ge1$, $P_n=\displaystyle\int_1^e(\log x)^n\,dx$, then $P_{10}-90P_8$ is equal to :

The value of the integral $\displaystyle\int_{-\pi/2}^{\pi/2}\left(x^2+\ln\frac{\pi+x}{\pi-x}\right)\cos x\,dx$ is

The area of the region bounded by the curve $y=f(x)$, the $x$-axis, and the lines $x=a$ and $x=b$, where $-\infty<a<b<-2$, is