$\int x \cos x \, dx = $

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.)

$\int e^x \cdot \cos 2x \, dx = $ . . . . . . $+ C$.

If $\int {x{e^{2x}}\,dx} $ is equal to ${e^{2x}}f(x) + C$,where $C$ is the constant of integration,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:

If $\int x^3(\log x)^2 d x = x^4[A(\log x)^2 + B(\log x) + C] + K$,then find the value of $A + B + C$.