In a classical mechanics text, I saw an integration(as part of a math) that uses gamma function as follows:
$$\int_0^{\pi/2} (\cos^2 x-\cos^3 x)dx$$ $$=\frac{\Gamma(\frac{2+1}{2})\Gamma(\frac{0+1}{2})}{2\Gamma(\frac{2+0+2}{2})}-\frac{\Gamma(\frac{3+1}{2})\Gamma(\frac{0+1}{2})}{2\Gamma(\frac{3+0+2}{2})} $$ What is the exact formula that is used here?
They are using the Beta function and in particular its defining property: $$\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}=B(x,y) = 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,\mathrm{d}\theta, \qquad \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0$$