A) \[\log \left( \log \tan \frac{x}{2} \right)+c\]
B) \[2\log \left( \log \tan \frac{x}{2} \right)+c\]
C) \[\frac{1}{2}\log \left( \log \tan \frac{x}{2} \right)+c\]
D) None of these
Correct Answer: A
Solution :
Put \[\log \tan \frac{x}{2}=t\Rightarrow \frac{1}{\tan \frac{x}{2}}.\frac{1}{2}{{\sec }^{2}}\frac{x}{2}\,dx=dt\] \[\Rightarrow \text{cosec}\,x\,dx=dt,\] therefore \[\int_{{}}^{{}}{\frac{\text{cosec}\,x}{\log \tan \frac{x}{2}}\,dx}=\int_{{}}^{{}}{\frac{1}{t}dt}=\log t+c=\log \left( \log \tan \frac{x}{2} \right)+c\].
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