Question

Integrate the rational function: \(\frac {2x}{(x^2+1)(x^2+3)}\)

Answer 1

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

\(Let  \ x^2 = t ⇒ 2x \ dx = dt\)

∴ \(∫\)\(\frac {2x}{(x^2+1)(x^2+3)}dx\) = \(∫\frac {dt}{(t+1)(t+3)}\)             ...(1)

\(Let\) \(\frac {dt}{(t+1)(t+3)}\) = \(\frac {A}{(t+1)}+\frac {B}{(t+3)}\)

\(1 = A(t+3)+B(t+1)\)                                              ...(1)

Substituting t = −3 and t = −1 in equation (1), we obtain

\(A = \frac 12\  and \ B = -\frac 12\)

∴ \(\frac {1}{(t+1)(t+3)}\) = \(\frac {1}{2(t+1)}-\frac {1}{2(t+3)}\)

⇒ \(∫\)\(\frac {2x}{(x^2+1)(x^2+3)}dx\) = \(∫\)\([\frac {1}{2(t+1)}-\frac {1}{2(t+3)}]dt\)

                                           =\(\frac 12log\ |(t+1)|-\frac 12log\ |t+3|+C\)

                                           =\(\frac 12\ log|\frac {t+1}{t+3}|+C\)

                                           =\(\frac 12\ log|\frac {x^2+1}{x^2+3}|+C\)