Solving Anti-Derivatives: The Case of (2 + x^2)/(1 + x^2)

[theRukus]

Homework Statement

Find the anti-derivative of (2 + x^2)/(1 + x^2)

Homework Equations

f(x) = tan^-1(x)
f'(x) 1/(1 + x^2)

The Attempt at a Solution

(2 + x^2) / (1 + x^2)

= ( 2 / (1 + x^2) ) + ( x^2 / (1 + x^2) )

The anti-derivative of (2 / (1 + x^2) ) is 2tan^-1(x). I don't know how to go about taking the anti-derivative of (x^2 / (1 + x^2) ). Could anyone give me a nudge in the right direction?


Thank you!

 

[Dick]

I think you should do polynomial division on (2+x^2)/(1+x^2) before you start integrating. It would help if you can show that's 1+1/(1+x^2), yes?

 

[Kevin_Axion]

Well the original integral $$\int\frac{(2+x^2)}{(1+x^2)}dx$$ can be rewritten as $$\int\frac{1}{x^2+1}+1 dx$$.
As referenced above: http://en.wikipedia.org/wiki/Polynomial_long_division.