Integrate the rational function: x (x+1)(x+2) - Vidyadip

Question

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

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Answer

Let $\frac{x}{(x+1)(x+2)}=\frac{A}{(x+1)}+\frac{B}{(x+2)}$
$\Rightarrow x = A(x + 2) + B(x + 1)$
On comparing the coefficients of x and constant term, we get,
$A + B = 1$
$2A + B = 0$
On solving above two equations, we get,
$A = -1$ and $B = 2$
Thus,
$\frac{x}{(x+1)(x+2)}=\frac{-1}{(x+1)}+\frac{2}{(x+2)}$
$\Rightarrow$ $\int \frac{x}{(x+1)(x+2)}=\int\left\{\frac{-1}{(x+1)}+\frac{2}{(x+2)}\right\} d x$
$= -\log|x + 1| + 2 \log|x + 2| + C$
$= \log (x + 2)^2 - \log |x + 1| + C$
$= \log \frac{(x+2)^{2}}{(x+1)}+C $

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