Question

Integrate the rational functions.
$\int \frac{2x}{(x^2+1)(x^2+3)}dx$.

Answer 1

Let $I=\int \frac{2x}{(x^2+1)(x^2+3)}dx$
Put $x^2=t\Rightarrow 2x=\frac{dt}{dx}\Rightarrow dx=\frac{dt}{2x}$
$\therefore I=\int \frac{2x}{(t+1)(t+3)}\frac{dt}{2x}=\int \frac{1}{(t+1)(t+3)}dt$
Let $\frac{1}{(t+1)(t+3)}=\frac{A}{t+1}+\frac{B}{t+3}=\frac{At+3A+Bt+B}{(t+1)(t+3)}$
$\Rightarrow 1=(A+B)t+(3A+B)$
On comparing the coefficients of t and constant terms on both sides, we get
A+B=0 .....(i)
and 3A+B =1 ....(ii)
On subtracting Eq. (ii)from Eq. (i), we get
$2A=1\Rightarrow A=\frac{1}{2}$
On putting the value of A in Eq. (i), we get $B=-\frac{1}{2}$
$\therefore I=\frac{1}{2}\int \frac{1}{t+1}dt-\frac{1}{2}\int \frac{1}{t+3}dt=\frac{1}{2}log|t+1|-\frac{1}{2}log|t+3|+C=\frac{1}{2}log\left|\frac{t+1}{t+3}\right|+C=\frac{1}{2}log\left|\frac{x^2+1}{x^2+3}\right|+C\text{[Put t=}{x}^2\text{]}$