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प्रश्न
` ∫ {"cosec" x }/ { log tan x/2 ` dx
योग
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उत्तर
` Note : "Here, we are considering " log x as log_e x `
\[\text{Let I} = \int\frac{cosec x}{\log \tan\frac{x}{2}}dx\]
\[Putting\ \log \tan \frac{x}{2} = t\]
\[ \Rightarrow \frac{1}{2}\frac{\sec^2 \frac{x}{2}}{\tan\frac{x}{2}} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{1}{2 \sin\frac{x}{2} . \cos\frac{x}{2}} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{1}{\sin x} = \frac{dt}{dx}\]
\[ \Rightarrow \text{cosec x dx} = dt\]
\[ \therefore I = \int\frac{dt}{t}\]
\[ = \text{log}\left| t \right| + C\]
\[ = \text{log }\left| \log \tan\frac{x}{2} \right| + C\]
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