Question

If $\int \frac{f\left(x\right)}{\log \left(\sin x\right)}dx=\log \left[\log \sin x\right]+c$, then $f\left(x\right)=$........

• A. $\cot x$
• B. $\tan x$
• C. $\sec x$
• D. $co\sec x$

Answer 1

Answer: (A)

$\cot x$

Here, $\int \frac{f\left(x\right)}{\log \left(\sin x\right)}dx=\log \left(\log \right(\sin x\left)\right)+c$

R.H.S.=$\log \left(\log \right(\sin x\left)\right)$

Differentiating the R.H.S

$\Rightarrow$ $\frac{1}{\log \left(\sin x\right)}\times \frac{1}{\sin x}\times \cos x=\frac{\cot x}{\log \left(\sin x\right)}$

$\Rightarrow f\left(x\right)=\cot x$