How do you integrate ln(x)?

Here, we use integration by parts. We must imagine ln(x) as a product of 1 and ln(x). We usually take the function of x to be our dv/dx, however, in the case of ln(x), we take that to be u (it is a special case) and dv/dx=1. Following the rule: int(1ln(x))dx = uv - int(vu')dx ... We achieve: = xln(x) - int(x/x)dx = xln(x) - x + c We must remember to add our constant of integration on the end as it is an indefinite integral. Our numerator within the integral, v, comes from integrating dv/dx=1, achieving v=x, and x/x=1, which integrates to x.