Question

Integrate $\frac{\cot \left(\log x\right)}{x}.$

• A. $\log \left(\sin \log x\right).$
• B. $\sin \log x$
• C. $\log \left(\log x\right).$
• D. $-\log \left(\sin \log x\right).$

Answer 1

The correct option is A $\log \left(\sin \log x\right).$

Let $I=\int \frac{\cot \left(\log x\right)}{x}dx$

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

$\therefore I=\int \cot tdt=\log \left(\sin t\right)+c=\log \left(\sin \left(\log x\right)\right)+c$