Question

Integrate the following using INFINITE SERIES (no binomial please)?

`\int(\ln(1+x^2))/x^wdx`

Answer 1

`int ln(1+x^2)/x^w = C+sum_(n=0)^oo (-1)^n/(n+1)x^(2n+3-w)/(2n+3-w)`

Explanation:

Using the MacLaurin expansion of:

`ln(1+t) = sum_(n=0)^oo (-1)^nt^(n+1)/(n+1)`

let: `t= x^2`: to get

`ln(1+x^2) = sum_(n=0)^oo (-1)^nx^(2n+2)/(n+1)`

then divide by `x^w` and integrate term by term:

`ln(1+x^2)/x^w = sum_(n=0)^oo (-1)^nx^(2n+2-w)/(n+1)`

`int ln(1+x^2)/x^w = sum_(n=0)^oo (-1)^n/(n+1) int x^(2n+2-w)dx`

`int ln(1+x^2)/x^w = C+sum_(n=0)^oo (-1)^n/(n+1)x^(2n+3-w)/(2n+3-w)`

The series has radius of convergence at least equal to the original series `R=1`, but has sense only if for every `n`:

`2n+3-w != 0`

`2n+3 != w`

thus `w` cannot be an odd integer number except for `w=1`.