Help to prove a reduction formula

[rock.freak667]

Homework Statement

Let [tex]I_n=\int_{0}^{1} (1+x^2)^{-n} dx[/tex] where [tex]n\geq1[/tex]
Prove that [tex]2nI_{n+1}=(2n-1)I_n+2^{-n}

Homework Equations

consider:
[tex] \frac{d}{dx}(x(1+x^2)^n) [/tex]

The Attempt at a Solution

[tex]\frac{d}{dx}(x(1+x^2)^n = (1+x^2)-2nx^2(1+x^2)^{-n-1}[/tex]

Integrating both sides between 1 and 0

[tex] \left[ x(1+x^2)^n \right]_{0}^{1} = I_n -2n\int_{0}^{1} x^2(1+x^2)^{-n-1}[/tex]

[tex]2n\int_{0}^{1} x^2(1+x^2)^{-n-1}[/tex] = [tex]\left[ \frac{x^2(1+x^2)^{-n-1}}{2} \right]_{0}^{1} + (n+1)\int_{0}^{1} x^3(1+x^2)^{-n-2}[/tex]

which is even more of story to integrate by parts..is there any easier way to integrate[tex]2nx^2(1+x^2)^{-n-1}[/tex] ?

 

[Avodyne]

Try expressing the integral of [tex]2nx^2(1+x^2)^{-n-1}[/tex] in terms of [tex]I_{n+1}[/tex] and [tex]I_{n}[/tex].