Answer
$$\int^4_1\frac{\log_{2}x}{x}dx=\ln 4$$
Work Step by Step
$$A=\int^4_1\frac{\log_{2}x}{x}dx=\int^4_1\frac{\frac{\ln x}{\ln2}}{x}dx$$ $$A=\frac{1}{\ln2}\int^4_1\frac{\ln x}{x}dx$$
We set $u=\ln x$, which means $$du=\frac{1}{x}dx$$
- For $x=4$, we have $u=\ln4$
- for $x=1$, we have $u=\ln1=0$
Therefore, $$A=\frac{1}{\ln2}\int^{\ln4}_0 udu$$ $$A=\frac{1}{\ln2}\times\frac{u^2}{2}\Big]^{\ln4}_0$$ $$A=\frac{1}{2\ln2}((\ln4)^2-0^2)$$ $$A=\frac{(\ln4)^2}{2\ln2}=\ln4$$
