Evaluate \[ \int_0^{\frac{1}{2}\pi} \cos^3 x dx, \quad \int_0^{\frac{1}{4}\pi} \frac{dx}{3+2\cos x}, \quad \int_0^1 \frac{x+2}{(4+x^2)(1-x)}dx. \]
Find the area between the curve \[ y^2(3a-x)=x^3 \] and its asymptote.