Reduction formulae

Question

$$I_{n}=\int x^n(1-x^2)^{\frac{1}{2}} dx$$ Show that $$(n+2)I_{n}=(n-1)I_{n-2}-x^{n-1}(1-x^2)^{\frac{3}{2}}$$

I can't seem to get this answer. Can someone please explain how to get to this?

Thanks in advance.

Answer 1

Rewrite $I_n$ in this way: $$ I_n = -\frac{1}{3}\int x^{n-1} \frac{3}{2}(1-x^2)^\frac{1}{2} (-2x) \, dx$$

and note that derivative of $(1-x^2)^{\frac{3}{2}}$ is $\frac{3}{2}(1-x^2)^\frac{1}{2} (-2x)$.

Then integrate by parts and rearrange.