I want to compute the following ratio: $$\frac{\int_0^1x^n\sqrt{1-x^2}}{\int_0^1x^{n-2}\sqrt{1-x^2}}$$
As I found from wolframe it is $\frac{n-1}{n+2}$. I am trying to solve the integral using the following substitution:
$$f(x) = x^{n-2}\sqrt{1-x^2}$$ $$\frac{\int_0^1x^2f(x)}{\int_0^1f(x)}$$
But I am sticking here.
$$x=\sin y $$ $$\implies \frac{\int_0^\frac{\pi}{2}\sin^n y\cos^2y\ dy }{\int_0^\frac{\pi}{2}\sin^{n-2}y\cos^2y\ dy}$$
Use the reduction rule or Wallis formula for cosine/ sine rule.
Note that the integral $$\int_0^1 x^{n} \sqrt{1-x^2} dx$$ is a Beta function so you can easily count it.
Otherwise, if you don't know the Beta function so mark $x=\sin{t}$ and again you'll easily count the integral.
