If $\int_0^a (x^2 - 4x + 1)dx = 6$, then the real value of $a$ is

If $\int_{\pi/6}^{\pi/3} \dfrac{1}{1+\sin x + \cos x}dx = \log 2$, then the value of $\int_{\pi/6}^{\pi/3} \dfrac{\cos x}{1+\sin x + \cos x}dx = $ ............

The area of the region bounded by the curve $y = 2^{kx}$ and $x = 0, x = 2$, in first quadrant is $\dfrac{2}{\log 2}$, then the value of $k$ is

The area of the region bounded by the curve $y = x^3$ and the lines $y = 8$ and $x = 0$ is ... square units.