If $f(x) = \int \operatorname{cosec}^5 x \, dx$,then $f\left(\frac{\pi}{4}\right) = $

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

If $\int(\sin x )^{\frac{-11}{2}}(\cos x )^{\frac{-5}{2}} dx = -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x)^{\frac{-3}{2}}+C,$ where $p_i$ and $q_i$ are positive integers with $\operatorname{gcd}(p_i, q_i)=1$ for $i =1,2,3,4$ and $C$ is the constant of integration, then $\frac{15 p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is equal to . . . . . . .

$\int \frac{d x}{\left(x^2-a^2\right)^{\frac{3}{2}}}$ is equal to

$\int \frac{25 x^2+8}{\sqrt{25 x^2+9}} d x=$

Let $I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x$ and $J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x$. Then,for an arbitrary constant $C$,the value of $J-I$ equals