What is the integral of ln(x) dz?

To find the integral of ln(x) with respect to x (as the original question intends to imply integration in the context of respect to x, not dz), we can apply integration by parts. The formula for integration by parts is given by:

[

\int u , dv = uv - \int v , du

]

In this case, we choose ( u = \ln(x) ) and ( dv = dx ). Consequently, we need to compute ( du ) and ( v ):

• From ( u = \ln(x) ), we have ( du = \frac{1}{x} , dx ).

• From ( dv = dx ), integrating gives us ( v = x ).

Substituting into the integration by parts formula results in:

[

\int \ln(x) , dx = x \ln(x) - \int x \left(\frac{1}{x}\right) , dx

]

This simplifies to:

[

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

]

where ( C ) is the constant of integration.