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∫ Tan 2 X Tan 3 X Tan 5 X D X - Mathematics

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Question

` ∫ tan 2x tan 3x tan 5x dx `

Sum

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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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Indefinite Integral Problems

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Chapter 19: Indefinite Integrals - Exercise 19.08 [Page 48]

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RD Sharma Mathematics [English] Class 12

Chapter 19 Indefinite Integrals
Exercise 19.08 | Q 46 | Page 48

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