$\int \frac{e^{2x} - 1}{e^{2x} + 1} dx =$

If $\dfrac{d}{dx}[f(x)] = 5x^4 - \dfrac{2}{x^3}$ such that $f\left(-\dfrac{1}{2}\right) = 0$, then $f(x)$ is equal to

For $x \in \left(0, \frac{\pi}{2}\right)$, $\int \frac{1}{\sin^2 x + \sin 2x} dx$ is equal to

$\int\frac{x-1}{x(x-\log x)}dx$ equals:

(consider: $\log x=\log_e x$)

$\displaystyle\int_0^2 x\sqrt{2-x}\,dx$ is equal to :

$\int \frac{x^2 + 1}{x^4 + 1} dx =$

$\int_0^{4\pi} \frac{x}{1 + |\cos x|} dx =$