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. $cosec\,x$

Answer 1

Answer: (A)

$cot\,x$

Answer 2

Given, $\int \frac{f(x)}{log (\sin x)} dx=\log [log\, sin x]+c$
On differentiating both sides, we get
$\frac{f(x)}{log (sin x)}=\frac{1}{log sin x} \frac{d}{dx}(log \,sin x)+0$
$\Rightarrow \frac{f(x)}{log (sin \,x)}=\frac{1}{log \,sin x} \times \frac{1}{sin\, x} \times \cos x$
$\Rightarrow f(x)=cot \,x$