Question

Integrate $\int \frac{2x}{(x^2+1)(x^4+4)}dx$.

Answer 1

Let $x^2=t$

$\Rightarrow 2xdx=dt$

$\Rightarrow I=\int \frac{dt}{(t+1)(t^2+4)}$

Let $\frac{1}{(t+1)(t^2+4)}=\frac{A}{t+1}+\frac{Bt+C}{t^2+4}$

$\Rightarrow \frac{1}{(t+1)(t^2+4)}=\frac{A{t}^2+4A+B{t}^2+Bt+Ct+C}{(t+1)(t^2+4)}$

$\Rightarrow A+B=0$; $C+B=0$; $C+4A=1$

$\Rightarrow A=-B$ $C=-B$ $\Rightarrow -5B=1$ $\Rightarrow B=\frac{-1}{5}$

$\Rightarrow A=\frac{1}{5}$, $c=\frac{1}{5}$

$\Rightarrow I=\frac{1}{5}\int \frac{dt}{t+1}-\frac{1}{5}\frac{t}{t^2+4}dt+\frac{1}{5}\int \frac{dt}{t^2+4}$

$=\frac{1}{5}ln|t+1|-\frac{1}{10}ln|t^2+4|+\frac{1}{5\left(2\right)}{\tan }^{-1}\left(\frac{t}{2}\right)+c$

$\Rightarrow I=\frac{1}{5}ln|x^2+1|-\frac{1}{10}ln|x^4+4|+\frac{1}{10}{\tan }^{-1}\left(\frac{x^2}{2}\right)+c$.