Calc1.611

From Example Problems
Jump to: navigation, search

\int {\frac  {\ln {x}}{x}}dx\,=?
In order to compute this integral, are going to integrate it by parts. Recall that the formula for this method is \int u\,du=uv-\int v\,du\,.
Let
u\,=\ln x\, and dv\,={\frac  {1}{x}}dx\,.
Then
du\,={\frac  {1}{x}}dx\, and v\,=\ln x\,.
Substituting that into the formula above, we get the following:

\int {{\frac  {\ln {x}}{x}}\,dx}\ =\ln ^{2}x-\int {{\frac  {\ln x}{x}}\,dx}

By simplifying the equation above, we get that

\int {\frac  {\ln {x}}{x}}dx={\frac  {\ln ^{2}x}{2}}+c.


Calculus
Main Page