Question

integrate the rational function 5x/(x+1) (x ^2-4) ​

Answer 1

Given :-

$\sf \dfrac{5x}{(x+1)(x^{2}-4)} =\dfrac{5x}{(x+1)(x+2)(x-2)}$

Solution :-

Also be,

$\sf 5x=A(x+2)(x-2)+B(x+1)(x-2)+C(x+1)(x+2) \quad ....(1)$

By substituting,

$\sf A=\dfrac{5}{3}$

$\sf B=\dfrac{-5}{2}$

$\sf C=\dfrac{5}{6}$

Now, we get

$\sf \dfrac{5x}{(x+1)(x+2)(x-1)} =\dfrac{5}{3(x+1)} -\dfrac{5}{2(x+2)} +\dfrac{5}{6(x-1)}$

By integrating,

$\sf \int \dfrac{5x}{(x+1)(x^{2}-4)} dx=\dfrac{5}{3} \int \dfrac{1}{(x+2)} dx-\dfrac{5}{2} \int \dfrac{1}{(x+2)} dx+\dfrac{5}{6} \int \dfrac{1}{(x-2)} dx$

$\sf =\dfrac{5}{3} log \right{|}x+1 \left{|}-\dfrac{5}{2} log \right{|}x-2 \left{|}+C$