Q11  Integrate the rational functions \frac{5x}{(x+1)(x^2-4)}

Answers (1)

Given function \frac{5x}{(x+1)(x^2-4)}

can be rewritten as \frac{5x}{(x+1)(x^2-4)} = \frac{5x}{(x+1)(x+2)(x-2)}

The partial function of this function:

\frac{5x }{(x+1)( x+2)(x-2)} = \frac{A}{(x+1)} +\frac{B}{(x+2)}+\frac{C}{(x-2)}

\Rightarrow (5x) =A(x+2)(x-2)+B(x+1)(x-2)+C(x+1)(x+2) 

Now, substituting the value of x =-1,-2,\ and\ 2 respectively in the equation above, we get

A=\frac{5}{3},\ B =\frac{-5}{2},\ and\ C= \frac{5}{6}

Therefore, 

\frac{5x }{(x+1)( x+2)(x-2)} = \frac{5}{3(x+1)} -\frac{5}{2(x+2)}+\frac{5}{6(x-2)}

\implies \int \frac{5x}{(x+1)(x^2-4)}dx = \frac{5}{3}\int \frac{1}{(x+1)}dx -\frac{5}{2}\int \frac{1}{x+2}dx+\frac{5}{6}\int \frac{1}{(x-2)}dx= \frac{5}{3}\log|x+1| -\frac{5}{2}\log|x+2| +\frac{5}{6}\log|x-2|+C

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