Evaluate 2 x+1 (x+1)(x-2) d x - Vidyadip

Question

Evaluate $\int \frac{2 x+1}{(x+1)(x-2)} d x$

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Answer

Let $I =\int \frac{2 x+1}{(x+1)(x-2)} d x$
Let $\frac{2 x+1}{(x+1)(x-2)}=\frac{ A }{x+1}+\frac{ B }{x-2}$
$\therefore 2 x +1= A ( x -2)+ B ( x +1) \ldots\text{(i)}$
Putting $x=-1$ in (i), we get
$ 2(-1)+1= A (-1-2)+ B (0)$
$\therefore-1=-3 A$
$\therefore A =\frac{1}{3} $
Putting $x=2$ in (i), we get
$ 2(2)+1= A (0)+ B (2+1)$
$\therefore 5=3 B $
$\therefore B =\frac{5}{3}$
$\therefore \frac{2 x+1}{(x+1)(x-2)}=\frac{\left(\frac{1}{3}\right)}{x+1}+\frac{\left(\frac{5}{3}\right)}{x-2}$
$\therefore I =\int\left(\frac{\left(\frac{1}{3}\right)}{x+1}+\frac{\left(\frac{5}{3}\right)}{x-2}\right) d x$
$=\frac{1}{3} \int \frac{1}{x+1} d x+\frac{5}{3} \int \frac{1}{x-2} d x$
$\therefore I =\frac{1}{3} \log |x+1|+\frac{5}{3} \log |x-2|+ c $

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