$\int {\frac{{{e^{5\log x}} - {e^{4\log x}}}}{{{e^{3\log x}} - {e^{2\log x}}}}\;dx} = $

$\int \sqrt{1 + \sin \frac{x}{2}} \, dx = $

$\int {\frac{{1 + {{\cos }^2}x}}{{{{\sin }^2}x}}} \,dx = $

Let $f(x) = \tan^{-1}\left(\frac{1+\cos x}{\sin x}\right)$ and $g(x) = \tan^{-1}\left(\frac{\sin x}{1-\cos x}\right)$. Then,evaluate $\int (f(x) + g(x)) \, dx$.

Consider the following Assertion $(A)$ and Reason $(R)$:
Assertion $(A)$: $\int \sqrt{x-3} \left(\sin^{-1}(\log x) + \cos^{-1}(\log x)\right) dx = \frac{\pi}{3}(x-3)^{3/2} + c$
Reason $(R)$: $\sin^{-1}(f(x)) + \cos^{-1}(f(x)) = \frac{\pi}{2}$ for $|f(x)| \le 1$
Choose the correct option:

$\int \frac{1}{\sqrt{2x-x^2}} dx = $ . . . . . . $+ C$.