Prove the reduction formula?

Question

How does one use integration to prove this reduction formula:
http://img257.imageshack.us/my.php?image=mathqo4.png

Answer 1

Use integration by parts with
u = (x^2 + a^2)^n, ... v = x
Use IN to mean "the integral of" since I can't do the sign here.

IN[u(dv/dx)]dx
= uv - IN[v(du/dx)]dx
= x(x^2 + a^2) - IN[2n(x^2)(x^2 + a^2)^(n-1)]dx
= x(x^2 + a^2) - IN[2n(x^2 + a^2)(x^2 + a^2)^(n-1)]dx + 2n(a^2)IN[(x^2+a^2)^(n-1)]dx

The middle term is 2n multiplied by the original integral because
(x^2+a^2)(x^2+a^2)^(n-1)
=(x^2+a^2)^n

Hence if we add it to both sides we get
(2n+1)IN[(x^2+a^2)]dx
= x(x^2 + a^2) + (2na^2)IN[(x^2+a^2)^(n-1)]dx

Now divide both sides by 2n+1 and we have the required result.

Answer 2

Make a variable substitution for (x² + a²)^n and get another integral only in (x² + a²)^(n-1) so that, after n iterations, the integral becomes 1 dx.


Doug