I have a problem that i don't understand what the solutions manual did. I am learning as much calculus 2 over the summer as i can.

$\displaystyle \int \frac {x^3}{x^2 + 1}\,dx$

Here is the part where i am confused..

$\displaystyle \int [ {x} - \frac {x}{x^2 + 1}]\,dx $

Then,

$\displaystyle \int {x} \,dx - \frac {1}{2} \int \frac {2x}{x^2 +1} \,dx $

Finally

$\displaystyle \frac {1}{2} \, {x^2} - \frac {1}{2} ln(x^2 + 1) + c$

(using long division)

I am not sure how they got the answer by the steps. The first step is the most confusing. the second step i can see how they got it. Third step is confusing on how they got

$\displaystyle \frac {1}{2} \, {x^2} $.

The rest of it i understand.

Can anyone explain the first step and the third one?