Question

If $\int \frac{f\left(x\right)dx}{\log \left(\sin x\right)}$ $=\log \left(\log \right(\sin x\left)\right)$ then $f\left(x\right)=$

• A. $sinx$
• B. $cosx$
• C. $\log \left(sinx\right)$
• D. $cotx$

Answer 1

The correct option is B $cosx$

$\int \frac{f\left(x\right)}{log\left(sinx\right)}dx=log\left[log\right(sinx\left)\right]$

on differentiating both side cort x we get

$\frac{d}{dx}\int \frac{f\left(x\right)}{log\left(sinx\right)}dx=\frac{d}{dx}\left(log\right(logsinx\left)\right)$

$\frac{f\left(x\right)}{log\left(sinx\right)}=\frac{cosx}{log\left(inx\right)}$ $[\therefore \frac{d}{dx}logk=\frac{1}{k}]$

$\left[f\right(x)=cosx]$