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Find the integral $\int \tan (5x) \tan (3x) \tan(2x) \ dx $ .

This question is posted in a maths group in Facebook. What way should we use to solve integral like this?

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Let $t_n = \tan(nx)$, we have $$t_5 = \frac{t_3 + t_2}{1 - t_3t_2} \iff t_5 - t_5t_3t_2 = t_3 + t_2 \implies t_5t_3 t_2 = t_5 - t_3 - t_2$$ This leads to

$$\begin{align}\int \tan(5x)\tan(3x)\tan(2x) dx &= \int \left(\tan(5x) - \tan(3x) - \tan(2x) \right)dx\\ &= \frac12 \log\cos(2x) + \frac13\log\cos(3x) - \frac15\log\cos(5x) + \text{ const. } \end{align}$$

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  • $\begingroup$ Well done! (+1) $\endgroup$ – Mark Viola Dec 13 '16 at 2:35
  • $\begingroup$ @achille hui Nice trick! (+1) $\endgroup$ – Olivier Oloa Dec 13 '16 at 10:28

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