Answer
$$\int^e_1\frac{2\ln10\log_{10}x}{x}dx=1$$
Work Step by Step
$$A=\int^e_1\frac{2\ln10\log_{10}x}{x}dx=\int^e_1\frac{2\ln10\times\frac{\ln x}{\ln10}}{x}dx$$ $$A=2\int^e_1\frac{\ln x}{x}dx$$
We set $u=\ln x$, which means $$du=\frac{1}{x}dx$$
- For $x=e$, we have $u=\ln e=1$
- for $x=1$, we have $u=\ln1=0$
Therefore, $$A=2\int^{1}_0 udu$$ $$A=2\times\frac{u^2}{2}\Big]^{1}_0=u^2\Big]^{1}_0$$ $$A=1^2-0^2=1$$
