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Question
` ∫ tan 2x tan 3x tan 5x dx `
Sum
Solution
\[We\ know\ that, \]
\[ \tan 5x = \tan \left( 2x + 3x \right)\]
\[ \Rightarrow \tan 5x = \frac{\tan 2x + \tan 3x}{1 - \tan 2x \tan 3x}\]
\[ \Rightarrow \tan 5x - \tan 2x \tan 3x \tan 5x = \tan 2x + \tan 3x\]
\[ \Rightarrow \tan 2x \tan 3x \tan 5x = \tan 5x - \tan 2x - \tan 3x\]
\[ \therefore \int\tan 2x \tan 3x \tan 5x = \int\left( \tan 5x - \tan 2x - \tan 3x \right)dx\]
\[ = \frac{1}{5} \ln \left| \sec 5x \right| - \frac{1}{2} \ln \left| \sec 2x \right| - \frac{1}{3} \ln \left| \sec 3x \right| + C\]
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