A recursive formula for a certain family of integrals

Question

Any tips on how to do this? Sorry about the title.

Let $$I_n = \int_0^1\frac{x^{2n+1}}{\sqrt{1-x^2}}\;dx.$$ Show that $$I_n = \frac{2n}{2n+1}I_{n-1}.$$

And I have tried integration by parts! – user85798 Sep 29 '13 at 02:00 — user85798, Sep 29 '13 at 02:00

score 1 · Answer 1 · answered Sep 29 '13 at 02:08

1

Let $x=\sin t$. Then integrate by parts. You should get your integral to be

$$I_n=\int\limits_0^{\pi /2} {{{\sin }^{2n + 1}}tdt} $$

You will have to use that $\cos ^2t=1-\sin ^2t$.

answered Sep 29 '13 at 02:08

Pedro

Thanks! I've managed it now. – user85798 Sep 29 '13 at 02:51
1

@Oliver Nice. Then you might be interested in this. – Pedro Sep 29 '13 at 03:02

1 Answers1