<s>[INST] Evaluate the sum $x + \frac{x^3}{3} + \frac{x^5}{5} + ... $ 
Evaluate the sum $$x + \frac{x^3}{3} + \frac{x^5}{5} + ... $$

I was able to notice that:  $$ \sum_{n=0}^\infty \frac{x^{2n-1}}{2n-1} = \sum_{n=0}^\infty \int x^{2n-2}dx = \lim_{N\to\infty} \sum_{n=0}^N \int x^{2n-2} dx  $$
Where should I take it from here? (assuming I'm not the right way)
EDIT
Following the anwer:
$$\int \sum_{n=0}^\infty x^{2n-1} dx = \int \frac{\sum_{n=0}^\infty (x^2)^n}{x} dx = \int \frac{\frac{1}{1-x^2}}{x} dx= \int \frac{1}{x(1-x^2)}dx = \int \frac{1-x^2+x^2}{x(1-x^2)} = \int \frac{1}{x} dx + \int \frac{x}{1-x^2}dx = \ln(x) + \int \frac{1}{2}\frac{1}{1-t}dt + C = \ln(x) - \frac{\ln(x^2)}{2} + C$$  


*

*Is that right?

*How to evaluate $C$?

 [/INST] Consider,
$$\sum_{n\ge 1} x^{n-1} = \frac{1}{1-x}$$
Let $x \rightarrow x^2$ and then integrate. 
</s>