Question

Integrate $\frac{2x}{(x^2+1)(x^2+2)}$ with respect to x.

Answer 1

$I=\int \frac{2x}{(x^2+1)(x^2+2)}$

Let's solve $\frac{1}{(x^2+1)(x^2+2)}=\frac{A}{x^2+1}+\frac{B}{x^2+2}$

$=\frac{A(x^2+2)+B(x^2+1)}{(x^2+1)(x^2+2)}$

$⟹A+B=0,2A+B=1$ ............ (Equating coefficients)

$⟹A=-B,-2B+B=1$

$⟹B=-1$ and $A=1$

$⟹I=\int (\frac{1}{(x^2+1)}-\frac{1}{(x^2+2)})2xdx$

$I=\int (\frac{2xdx}{(x^2+1)}-\frac{2xdx}{(x^2+2)})$

$=\ln |x^2+1|-\ln |x^2+2|=\ln \frac{|x^2+1|}{|x^2+2|}$

Hence, $I=\int \frac{2x}{(x^2+1)(x^2+2)}=\ln \left(\frac{|x^2+1|}{|x^2+2|}\right)$