Inte-great!

Calculus Level 2

$\Large \int _{ 1 }^{ e }{ \cfrac { \ln { x } }{ x } } \, dx = \, ?$

Bonus: Find the value of the integral in two different ways.

2 solutions

Sam Bealing
May 8, 2016

$u=\ln(x) \Rightarrow \dfrac{du}{dx}=1/x \Rightarrow x \cdot du=dx$

$\int_{1}^{e} \dfrac{\ln(x)}{x} dx=\int_{0}^{1} u \cdot du=\left [\dfrac{u^2}{2} \right]_{0}^{1}=\boxed{0.5}$

Simple standard approach.

Did the same. Easiest method for this ;)

Nihar Mahajan - 5 years, 1 month ago

Yup. Did in the same manner. !!!Substitution. By the way nice and clear sol.+1

Rishabh Tiwari - 5 years, 1 month ago

Michael Fuller
May 8, 2016

Using integration by parts with $u=\ln x$ and $v'=x^{-1}$ :

$\int_{1}^{e} \dfrac{\ln x}{x} \, dx= [\ln ^2 x]_{1}^{e} - \int_{1}^{e} \dfrac{\ln x}{x} \, dx$

$\Rightarrow 2 \int_{1}^{e} \dfrac{\ln x}{x} \, dx= 1$

$\Rightarrow \int_{1}^{e} \dfrac{\ln x}{x} \, dx= \large \color{#20A900}{\boxed{0.5}}$

Parts is nice too. +1

Rishabh Tiwari - 5 years, 1 month ago