Complex definite Integration

Question

Actually i don't know how to start with this problem. I have taken many substitutions but its becoming more and more complex. Please guide me to solve it. Thanks

Answer 1

Trick: substitute $t = 1/x$, rename $t$ as $x$ and then add the result to this.

Answer 2

Taking $u=\dfrac{1}{x}$, The integration changes to $$\int_{{2}}^{\frac{1}{2}}\frac{u\sin(\frac{1}{u})}{\sin(\frac{1}{u})+\sin(u)}\frac{-1}{u^2}du=\int_{\frac{1}{2}}^{2}\frac{\sin(\frac{1}{u})}{u(\sin(u)+\sin\frac{1}{u})}du=\int_{\frac{1}{2}}^{2}\frac{\sin(\frac{1}{x})}{x(\sin(x)+\sin\frac{1}{x})}dx$$

Adding the two things we get $$2I=\int_{\frac{1}{2}}^{2}\frac{1}{x}dx$$

And we are done.