Solving this definite integral using integration by parts

[songoku]

Homework Statement

Please see below

Relevant Equations

Integration

Using integration by parts:
$$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$
$$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$
Then how to continue?

Thanks

 

[scottdave]

My first approach would be work out what I_n+1 is and see how they relate.

Their hint of multiplying by 1 tells me that I might need to represent 1 as a fraction where numerator = denominator.

 

[vela]

Hint: $x^2 = (x^2+1) - 1$.

 

[songoku]

I understand

Thank you very much scottdave and vela