Find the anti derivative of f'(x)= (2+x^2)/(1+x^2)

Homework Statement

f'(x)= (2+x^2)/(1+x^2), find f(x).

The Attempt at a Solution

I am pretty sure you have to use integration by parts... but I don't think we learned it yet. Is there a way to do this without using integration by parts? If not, how would I use integration by parts for his question?

gb7nash
Homework Helper

Homework Statement

f'(x)= (2+x^2)/(1+x^2), find f(x).

The Attempt at a Solution

I am pretty sure you have to use integration by parts... but I don't think we learned it yet. Is there a way to do this without using integration by parts? If not, how would I use integration by parts for his question?

Well, $\frac{2+x^2}{1+x^2} = \frac{2}{1+x^2}+\frac{x^2}{1+x^2}$

If you know what the integral of $\frac{1}{1+x^2}$ is, the first part is easy. The second part is simple as well, if you recognize what to do first before integrating.

Ray Vickson
Well, $\frac{2+x^2}{1+x^2} = \frac{2}{1+x^2}+\frac{x^2}{1+x^2}$
If you know what the integral of $\frac{1}{1+x^2}$ is, the first part is easy. The second part is simple as well, if you recognize what to do first before integrating.
You could also write $$\frac{2+x^2}{1+x^2} = 1 + \frac{1}{1+x^2}.$$