Given that $\int_0^\infty \frac{x^2 \, dx}{(x^2 + a^2)(x^2 + b^2)(x^2 + c^2)} = \frac{\pi}{2(a + b)(b + c)(c + a)}$,then the value of $\int_0^\infty \frac{x^2 \, dx}{(x^2 + 4)(x^2 + 9)}$ is

If $\int \frac{dx}{(x \tan x + 1)^2} = f(x) + c$,then $\lim_{x \rightarrow \frac{\pi}{2}} f(x) = $

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

$\int \frac{(1-4 \sin^2 x) \cos x}{\cos (3x+2)} dx =$

If $\int\left(\frac{1}{x}+\frac{1}{x^3}\right)\left(\sqrt[23]{3 x^{-24}+x^{-26}}\right) d x =-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^\gamma\right)^{\frac{\alpha+1}{\alpha}}+C, x>0,$ $(\alpha, \beta, \gamma \in Z)$,where $C$ is the constant of integration,then $\alpha+\beta+\gamma$ is equal to . . . . . . .

$\int \left( \frac{x}{x \cos x - \sin x} \right)^2 dx = $