Reduction Formulae: Defining I(n) & Establishing Reduction Formula

[Badrakhandama]

Define I(n) by I(n) = integral of x^n / square root(1+x^2) dx. Evaluate I(0), I(1) and then establish the reduction formula I(n) = ((root2)-(n-1)I(n-2))/(n) for suitable values of n, which should be stated




Here is my attempt


I found I(0) first,and got the answer to be ln(1+root2), and I(1) to be ln (root2).

For the reduction formula, I am not sure where to start - do I try it by parts, differentiating x^n, and integrating 1/(root(1+x^2)) , or what should I do? For the values of n, I am not sure either.

 

[SammyS]

Try the following for integration by parts & see where it leads.

$$\text{Let }\ dv=\frac{x}{\sqrt{1+x^2}}\,dx\ \text{ and let }\ u=x^{n-1}.$$

That should result in an integral with $$\sqrt{1+x^2}$$ in the numerator.

Multiply the integrand by $$\frac{\sqrt{1+x^2}}{\sqrt{1+x^2}}\ ,$$ and simplify the numerator.