Q: $\displaystyle I_n(x) = \displaystyle\int^x_0 \tan ^x \theta \, \mathrm{d}\theta, \ \ n \ge 0, |x|<\frac{\pi}{2}$

By writing $\displaystyle \tan \theta$ as $\displaystyle \tan ^{n-2} \theta \tan ^2 \theta$, or otherwise, show that

$\displaystyle I_x(x) = \frac{1}{n-1} \tan ^{n-1}x - I_{n-2}(x), \ \ n \ge 2, |x| <\frac{\pi}{2}$

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Help? Which to integrate and differentiate and when you do these what do you get? Thanks in advance. (Smile)