How to integrate ln(x)

How to integrate ln(x)?

In order to integrate log x of base e, we are going to apply integration by parts.

Recall that the formula for integration by parts is:

  /                                  /

 | f(x) g'(x) = f(x) g(x) - | f'(x) g(x)

 /                                 /

The application of integration by parts is interesting because there is only one function being integrated. We need an f and g'. The key step in this problem is we can manufacture a function by making

ln(x) = 1 * ln (x)

We can choose f(x) = ln (x) , g'(x) = 1 ==>>>> f'(x) = 1/x, g(x) = x

Then,

 /            /                                  /                                    /

 | ln(x) = | 1ln(x) dx = xln(x) - | x * (1/x) dx = x*ln(x) - | 1 dx 

/            /                                  /                                    /

 

    = x*ln(x) - x + C