Integrating $\int{\frac{2\,dx}{x^4+2x^2}}$

Question

I am stuck on this problem.

I proceeded to cheat and used a calculator, but got stuck again on this step:

How do we get from the first integration to the second ?

Answer 1

You have to put $$\dfrac{1}{x^4+2x^2}=\dfrac{1}{x^2(x^2+2)}=\dfrac{A}{x}+\dfrac{B}{x^2}+\dfrac{Cx+D}{x^2+2}$$ and then make the sum of fractions and compare to find $A,B,C,D$.

I haven´t checked, but given the answer probably you will get $A=0,B=\frac{1}{2},C=0,D=-\frac{1}{2}$.

Answer 2

It will be easier to solve if you substitute $x=\sqrt 2 \tan y$