$\int \frac{dx}{a^2 \sin^2 x + b^2 \cos^2 x}$ is equal to

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

Let $I(x) = \int \frac{x^2(x \sec^2 x + \tan x)}{(x \tan x + 1)^2} dx$. If $I(0) = 0$,then $I(\frac{\pi}{4})$ is equal to:

For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$,if $\int \left( \left( \frac{x}{e} \right)^{2x} + \left( \frac{e}{x} \right)^{2x} \right) \log_{e} x \, dx = \frac{1}{\alpha} \left( \frac{x}{e} \right)^{\beta x} - \frac{1}{\gamma} \left( \frac{e}{x} \right)^{\delta x} + C$,where $e = \sum_{n=0}^{\infty} \frac{1}{n!}$ and $C$ is the constant of integration,then $\alpha + 2\beta + 3\gamma - 4\delta$ is equal to:

If $\int \frac{d \theta}{\cos ^2 \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{e}|f(\theta)|+c$ (where $c$ is a constant of integration),then the ordered pair $(\lambda, |f(\theta)|)$ is equal to

For real numbers $\alpha, \beta, \gamma$ and $\delta,$ if $\int \frac{\left(x^{2}-1\right)+\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)}{\left(x^{4}+3 x^{2}+1\right) \tan ^{-1}\left(\frac{x^{2}+1}{x}\right)} d x =\alpha \log _{e}\left(\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)\right) +\beta \tan ^{-1}\left(\frac{\gamma\left(x^{2}-1\right)}{x}\right)+\delta \tan ^{-1}\left(\frac{x^{2}+1}{x}\right)+C$ where $C$ is an arbitrary constant,then the value of $10(\alpha+\beta \gamma+\delta)$ is equal to ....... .