The value of integral $\int_0^1\cot^{-1}(1 + x^2 - x)\,dx$ is...

The value of $\int_{-\pi}^{\pi}\dfrac{2x(1 + \sin x)}{1 + \cos^2 x}dx$ is...

$\displaystyle\int_{\pi/6}^{\pi/3} \dfrac{\sin x - \cos x}{1 + \sin x\cos x}\,dx =$

$\displaystyle\int_0^{\log 5} \dfrac{e^x\sqrt{e^x - 1}}{e^x + 3}\,dx =$