_2^3 d x x^2-x = - Vidyadip

MCQ

$\int_2^3 \frac{d x}{x^2-x}=$

A
$\log \left(\frac{2}{3}\right)$
B
$\log \left(\frac{1}{4}\right)$
✓
$\log \left(\frac{4}{3}\right)$
D
$\log \left(\frac{8}{3}\right)$

✓

Answer

Correct option: C.

$\log \left(\frac{4}{3}\right)$

(C)
$\int_2^3 \frac{d x}{x^2-x}=\int_2^3 \frac{d x}{x(x-1)}=\int_2^3\left[\frac{1}{x-1}-\frac{1}{x}\right] d x$
$\begin{array}{l}=\int_2^3 \frac{1}{(x-1)} d x-\int_2^3 \frac{1}{x} d x \\ =[\log (x-1)]_2^3-[\log x]_2^3 \\ =(\log 2-\log 1)-(\log 3-\log 2)=2 \log 2-\log 3 \\ =\log \left(\frac{4}{3}\right)\end{array}$

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