http://i41.tinypic.com/ml7ez6.jpg
Help please.
http://i41.tinypic.com/ml7ez6.jpg
Help please.
First off, there's a typo (an n is missing)
$\displaystyle \int(x^2+a^2)^ndx$ if $\displaystyle u = (x^2+a^2)^n,\;\; dv= dx$ so $\displaystyle du = 2 n x (x^2+a^2)^{n-1},\;\;v = x$ so
$\displaystyle I_n = \int(x^2+a^2)^ndx$
$\displaystyle = x(x^2+a^2)^n - 2n \int x^2 (x^2+a^2)^{n-1}dx$
$\displaystyle = x(x^2+a^2)^n - 2n \int (x^2+a^2-a^2) (x^2+a^2)^{n-1}dx$
$\displaystyle = x(x^2+a^2)^n - 2n \int (x^2+a^2) (x^2+a^2)^{n-1}dx + 2na^2 \int (x^2+a^2)^{n-1}dx$
$\displaystyle I_n = x(x^2+a^2)^n - 2n \underbrace{\int (x^2+a^2)^{n}dx}_{I_n} + 2na^2 \underbrace{\int (x^2+a^2)^{n-1}dx}_{I_{n-1}}$
so
$\displaystyle (1+2n) I_n = x(x^2+a^2)^n +2na^2 I_{n-1}$
from we obtain your result.