The value of $\int_0^{\pi/2} \log_e \left(\frac{5 + 2 \sin x}{5 + 2 \cos x}\right) dx$ is

$\int \frac{x}{(x-1)(x-2)} dx =$

(where c is an arbitrary constant)

Value of $\int (e^{2\log x} + e^{x\log 2} + e^{2\log 2})dx$ is

The value of $\int_0^{\pi/2} \frac{\cos x}{\cos\frac{x}{2}+\sin\frac{x}{2}}\,dx$ is

The value of $\int_{0}^{1} \frac{e^x}{1+e^{2x}}\,dx$ is

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{dx}{1 + \sqrt{\tan x}}$ is equal to :

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