<s>[INST] Evaluate $\int_\frac12 ^2 \frac1x\tan(x-\frac1x)dx $ $$\int\limits_\frac{1}{2} ^2 \frac{1}{x}\tan\left(x-\frac{1}{x}\right)\mathrm{d}x$$
I have tried substitution and by parts and it seems failed at all. Can anyone give me some hints?
 [/INST] $$I = \int_{\frac{1}{2}}^{2}\frac{1}{x}\tan\left(x - \frac{1}{x}\right)dx = \int_{2}^{\frac{1}{2}}u\cdot \frac{-1}{u^2}\tan\left(\frac{1}{u}-u\right)du = -\int_{\frac{1}{2}}^{2}\frac{1}{u}\tan\left(u - \frac{1}{u}\right)du = -I$$
Where the second step was obtained by letting $\displaystyle x = \frac{1}{u}, dx = \frac{-1}{u^2}du$.
Since $I = -I$, we have $I = 0$.
</s>