If $f(x) = \frac{1}{\log x}$ and $g(x) = \frac{1}{(\log x)^2}$,then evaluate the integral $\int \{f(x) - g(x)\} dx$. (Where $C$ is a constant of integration.)

If $\int x^3 e^{2 x} d x = \frac{e^{2 x}}{8} f(x) + c$,then the sum of all the complex roots of $f(x) = 1$ is

Integrate the function: $x \sin 3x$.

If $\int \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x=f(x)-\log \left(1+x^2\right)$,then $f(x)$ is equal to

If $I = \int \frac{x^2 \, dx}{(x \sin x + \cos x)^2} = f(x) + \tan x + c$,then $f(x)$ is

Let $\int x^3 \sin x \, dx = g(x) + C$,where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\right) + g^{\prime}\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma$,where $\alpha, \beta, \gamma \in \mathbb{Z}$,then $\alpha + \beta - \gamma$ equals: