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...

The area of the region enclosed by the lines $y = 2x, 2y = x$ and $x = 2$ (in sq.units) is...

The area of the region common to the parabolas $4y^2 = 9x$ and $3x^2 = 16y$ is...