Question

How to integrate ∫㏑x/χ
from 1 to e

Answer 1

`\int_1^e(\ln x)/(x)dx,~\text{if}~u=\ln x\to x=e^u\to dx=e^udu\\\\\text{So, finding the new limits:}\begin{cases}x=1\to u=\ln1\to u=0\\x=e\to u=\ln e\to u=1\end{cases}\\\\ \int_1^e(\ln x)/(x)dx=\int_0^1(u)/(e^u)e^udu=\int_0^1u\,du=\left[(u^2)/(2)\right]_0^1=(1^2)/(2)-(0^2)/(2)=(1)/(2)\\\\ \boxed{\int_1^e(\ln x)/(x)dx=(1)/(2)}`