Evaluate _2^3 x (x+2)(x+3) d x - Vidyadip

Question

Evaluate $\int_2^3 \frac{x}{(x+2)(x+3)} d x$

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Answer

Let $I =\int_2^3 \frac{x}{(x+2)(x+3)} d x$
Let $\frac{x}{(x+2) x+3}=\frac{ A }{x+2}+\frac{ B }{x+3}$ $\ldots \text{(i)}$
$\therefore x = A ( x +3)+ B ( x +2) \ldots \text{(ii)}$
Putting $x=-3$ in (ii), we get
$ -3=-B$
$\therefore B=3 $
Putting $x=-2$ in (ii), we get
$ -2=A$
$\therefore A=-2 $
From (i), we get
$\frac{x}{(x+2)(x+3)}=\frac{-2}{x+2}+\frac{3}{x+3}$
$\therefore I =\int_2^3\left[\frac{-2}{x+2}+\frac{3}{x+3}\right] d x$
$=-2 \int_2^3 \frac{1}{x+2} d x+3 \int_2^3 \frac{1}{x+3} d x$
$=-2[\log |x+2|]_2^3+3[\log |x+3|]_2^3$
$=-2(\log 5-\log 4)+3(\log 6-\log 5)$
$=-2 \log \left(\frac{5}{4}\right)+3 \log \left(\frac{6}{5}\right)$
$=3 \log \left(\frac{6}{5}\right){-2 \log \left(\frac{5}{4}\right)}$
$=\log \left(\frac{6}{5}\right)^3-\log \left(\frac{5}{4}\right)^2$
$=\log \left(\frac{216}{125}\right)-\log \left(\frac{25}{16}\right)$
$=\log \left(\frac{216}{125} \times \frac{16}{25}\right)$
$\therefore I =\log \left(\frac{3456}{3125}\right)$

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