If $\int \sqrt{x-\frac{1}{x}}\left(\frac{x^{2}+1}{x^{2}}\right) d x=\frac{2}{3}\left(x-\frac{1}{x}\right)^{k}+c$,then the value of $k$ is

If $\int \frac{e^x}{\sqrt{e^{2x}+4e^x+13}} dx = \log \left|e^x+2+\sqrt{e^{2x}+4e^x+13}\right|+c$,(where $c$ is the constant of integration),then the value of $a$ in the expression $\log \left|e^{ax}+2+\sqrt{e^{2x}+4e^x+13}\right|+c$ is equal to

Integrate the function $\frac{\sin x}{1+\cos x}$.

$\int \frac{\operatorname{cosec} x \, dx}{\cos ^2\left(1+\log \tan \frac{x}{2}\right)} = $

If $\int \frac{(x-1) dx}{(x+1) \sqrt{x^3+x^2+x}} = f(x) + C$, then $f(1) =$

If $\int \frac{\log _e(x+\sqrt{1+x^2})}{\sqrt{1+x^2}} dx = f(g(x)) + c$, then: