If $\int \frac{dx}{x^{7/2}(x^4+1)^{3/8}} = m \left(\frac{x^4+1}{x^4}\right)^n + c$, where $c$ is a constant of integration, then the value of $n$ is...

If $\int \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} = (\tan x)^A + C(\tan x)^B + k$ where $k$ is a constant of integration,then $A+B+C$ equals

If $\int f(x) \sin x \cos x \, dx = \frac{1}{2(b^2 - a^2)} \log f(x) + c$,where $c$ is the constant of integration,then $f(x)$ is

If $\int \frac{2 \, dx}{\sqrt{\cot^2 x - \tan^2 x}} = -\sqrt{f(x)} + c$, then $f(x) =$

For $x>0$, evaluate the integral: $\int \left( \frac{\sqrt{1+x+x^2}}{1+x} + \frac{1}{2 \sqrt{1+x+x^2}} - \frac{1}{(1+x) \sqrt{1+x+x^2}} \right) dx$.

$\int {\left( {\sin \left( {101x} \right).{{\sin }^{99}}x} \right)} dx = \frac{{\sin \left( {100x} \right){{\left( {\sin x} \right)}^\lambda }}}{\mu } + C$ where $C$ is the constant of integration,then $\frac{\lambda }{\mu }$ is equal to