How do you integrate (2x)/(1+x^2) with respect to x?

The key here is to recognise that this is in the form f'(x)/f(x). We can use the idea that integration is the inverse of differentiation, and the knowledge that the derivative of ln(f(x)) is equal to f'(x)/f(x). In this case f(x)=1+x^2, so we have that the integral of (2x)/(1+x^2) is equal to ln(1+x^2)+c.