This integral appeared in my exam but I couldn't solve it within the time limits (~2 minutes) $$I = \int\left[\left(\frac{x}{2}\right)^x + \left(\frac{2}{x}\right)^x\right]\log_2x \mathrm dx$$
A) $\left(\frac{x}{2}\right)^x + \left(\frac{2}{x}\right)^x$
B) $\left(\frac{x}{2}\right)^x -\left(\frac{2}{x}\right)^x$
C) $\left(\frac{x}{2}\right)^x\log_2x$
D) $\left(\frac{x}{2}\right)^x\log_x2$
E) $\left(\frac{2}{x}\right)^x\log_x2$
F) $\left(\frac{2}{x}\right)^x\log_2x$
I observed that the first two terms in addition are reciprocal. However, I got failed by trying substitution techniques ( like $t = (\frac{x}{2})^x$) and integration by parts is making it more messy. Can you give a hint/solution (like a proper substitution or method) to solve this ? Thanks !