The value of $\int_{4}^{7} \frac{(11-x)^{2}}{x^{2}+(11-x)^{2}} d x$ is

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ is equal to . . . . . . .

Match the following:

List-$I$	List-$II$
$I. \int_{-1}^1 x|x| dx$	$(a) \frac{\pi}{2}$
$II. \int_0^{\pi/2} \left(1 + \log \left(\frac{4+3\sin x}{4+3\cos x}\right)\right) dx$	$(b) \int_0^a 2f(x) dx$
$III. \int_0^a f(x) dx$	$(c) \int_0^a [f(x) + f(-x)] dx$
$IV. \int_{-a}^a f(x) dx$	$(d) 0$
	$(e) \int_0^a f(a-x) dx$

Let $I_{n} = \int_{1}^{e} x^{19}(\log |x|)^{n} dx$,where $n \in N$. If $(20) I_{10} = \alpha I_{9} + \beta I_{8}$,for natural numbers $\alpha$ and $\beta$,then $\alpha - \beta$ is equal to ..... .

If $A=\int_0^{\infty} \frac{1+x^2}{1+x^4} d x$ and $B=\int_0^1 \frac{1+x^2}{1+x^4} d x$,then

The value of $\int_{0}^{4042} \frac{\sqrt{x} \, dx}{\sqrt{x}+\sqrt{4042-x}}$ is equal to