What is the integral of ln(x) dx?

To find the integral of ( \ln(x) , dx ), we can use integration by parts. This technique is particularly useful when integrating the product of two functions, and in this case, we can choose:

• Let ( u = \ln(x) ) so that ( du = \frac{1}{x} , dx ).

• Let ( dv = dx ) so that ( v = x ).

Using the integration by parts formula, which states that ( \int u , dv = uv - \int v , du ), we can proceed with the substitution:

1. Calculate ( uv ):

[

uv = x \ln(x)

]

2. Calculate ( \int v , du ):

[

\int v , du = \int x \cdot \frac{1}{x} , dx = \int 1 , dx = x

]

Putting it all together:

[

\int \ln(x) , dx = x \ln(x) - \int 1 , dx = x \ln(x) - x + C

]

Thus, the final result of the integral is