Question

` ∫ {"cosec"   x }/ { log  tan   x/2 ` dx 

Answer 1

  ` Note : "Here, we are considering "  log x  as  log_e x `
\[\text{Let I} = \int\frac{cosec       x}{\log \tan\frac{x}{2}}dx\]
\[Putting\ \log \tan \frac{x}{2} = t\]
\[ \Rightarrow \frac{1}{2}\frac{\sec^2 \frac{x}{2}}{\tan\frac{x}{2}} = \frac{dt}{dx}\]

\[ \Rightarrow \frac{1}{2 \sin\frac{x}{2} . \cos\frac{x}{2}} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{1}{\sin x} = \frac{dt}{dx}\]
\[ \Rightarrow \text{cosec x dx} = dt\]
\[ \therefore I = \int\frac{dt}{t}\]
\[ = \text{log}\left| t \right| + C\]
\[ = \text{log }\left| \log \tan\frac{x}{2} \right| + C\]