Primitive function/Logarithm/Example

We determine a primitive function for the natural logarithm ${}\ln x$, with integration by parts. We write ${}\ln x=1\cdot \ln x$, and we integrate the constant function ${}1$, and we differentiate the logarithm. Then

${}\int _{a}^{b}\ln x\,dx=(x\cdot \ln x)|_{a}^{b}-\int _{a}^{b}x\cdot {\frac {1}{x}}\,dx=(x\cdot \ln x)|_{a}^{b}-\int _{a}^{b}1\,dx=(x\cdot \ln x)|_{a}^{b}-x|_{a}^{b}\,.$

So a primitive function is ${}x\cdot \ln x-x$.