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

[skyturnred]

Homework Statement

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

Homework Equations

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 Statement

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

Homework Equations

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}.

RGV

 

[skyturnred]

Thank you to both of you for your help, both of you helped me solve this and other similar questions.