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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