Help to prove a reduction formula

[rock.freak667]

Homework Statement

Let I_n=\int_{0}^{1} (1+x^2)^{-n} dx where n\geq1
Prove that 2nI_{n+1}=(2n-1)I_n+2^{-n}<h2>Homework Equations</h2><br /> <br /> consider:<br /> \frac{d}{dx}(x(1+x^2)^n)<h2>The Attempt at a Solution</h2><br /> <br /> \frac{d}{dx}(x(1+x^2)^n = (1+x^2)-2nx^2(1+x^2)^{-n-1}<br /> <br /> Integrating both sides between 1 and 0<br /> <br /> \left[ x(1+x^2)^n \right]_{0}^{1} = I_n -2n\int_{0}^{1} x^2(1+x^2)^{-n-1}<br /> <br /> 2n\int_{0}^{1} x^2(1+x^2)^{-n-1} = \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}<br /> <br /> which is even more of story to integrate by parts..is there any easier way to integrate2nx^2(1+x^2)^{-n-1} ?

 

[Avodyne]

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