Question

Integrate the given function $\frac{2x}{(x^2+1)(x^2+3)}$=

Answer 1

$\int \frac{2x}{(x^2+1)(x^2+3)}dx$

Take $x^2+1=t\Rightarrow 2xdx=dt$

$\Rightarrow \int \frac{dt}{t(t+2)}$

$\Rightarrow \int (\frac{1}{2t}-\frac{1}{2(t+2)})dt$

$\Rightarrow \int \frac{1}{2t}dt-\int \frac{1}{2(t+2)}dt$

$\Rightarrow \frac{1}{2}\log t-\frac{1}{2}\log (t+2)+C$

$\Rightarrow \frac{1}{2}\log \left(\frac{t}{t+2}\right)+C$

$\Rightarrow \frac{1}{2}\log \left(\frac{x^2+1}{x^2+3}\right)+C$