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

If $\int \cos x \log \left(\tan \frac{x}{2}\right) dx = \sin x \log \left(\tan \frac{x}{2}\right) + f(x)$, then $f(x)$ is equal to (assuming $c$ is an arbitrary real constant).

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 $A + B + C$.

If linear functions $f(x)$ and $g(x)$ satisfy $\int {\left[ {\left( {1 - 2x} \right)\cos x + \left( {3 + 2x} \right)\sin x} \right]} dx = f(x)\sin x + g(x)\cos x + C$ (where $C$ is the constant of integration),then which of the following is true?

Evaluate the integral: $\int \operatorname{Tan}^{-1}\left(x^{\frac{1}{3}}\right) d x$

If $\int {{x^5}{e^{ - 4{x^3}}}\,dx = \frac{1}{{48}}{e^{ - 4{x^3}}}f\left( x \right) + C} $,where $C$ is a constant of integration,then $f(x)$ is equal to