Question

Solve:
$\int \frac{2x}{(x^2+1)(x^2+2)}dx$

Answer 1

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

Let $x^2+1=t$

$\Rightarrow 2xdx=dt$

Therefore,

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

$=\int \frac{dt}{t(t+1)}$

$=\int \frac{dt}{t^2+t}$

$=\int \frac{dt}{{(t+\frac{1}{2})}^2-{\left(\frac{1}{2}\right)}^2}$

By using $\int \frac{1}{a^2-x^2}dx=\frac{1}{2a}\log |\frac{a+x}{a-x}|+c$

$=\frac{1}{2\times \left(\frac{1}{2}\right)}\ln \left|\frac{(t+\frac{1}{2})-\frac{1}{2}}{(t+\frac{1}{2})+\frac{1}{2}}\right|+C$

$=\ln \left|\frac{t}{t+1}\right|+C$

$=\ln \left|\frac{x^2+1}{x^2+2}\right|+C(\because t=x^2+1)$