Question

$\int \frac{dx}{x(\log x-2)(\log x-3)}=$

• A. $\frac{1}{x}\log \left|\frac{\log x-3}{\log x-2}\right|$
• B. $\log \left|\frac{\log x-3}{\log x-2}\right|$
• C. $\log \left|\frac{\log x-2}{\log x-3}\right|$
• D. $\log |(\log x-3)(\log x-2)|$

Answer 1

The correct option is B $\log \left|\frac{\log x-3}{\log x-2}\right|$

Let $I=\int \frac{dx}{x(\log x-2)(\log x-3)}$

Substitute $\log x=t\Rightarrow \frac{1}{x}dx=dt$

$\therefore I=\int \frac{dt}{(t-2)(t-3)}$ $=\int \frac{\left[\right(t-2)-(t-3\left)\right]dt}{(t-2)(t-3)}$

$=\log |t-3|-\ln |t-2|=\log \left|\frac{t-3}{t-2}\right|=\log \left|\frac{\log x-3}{\log x-2}\right|$