To find the integral of $\ln(x) \, dx$, we apply the method of integration by parts. The result of this integral is given by:

where $C$ represents the constant of integration.

We start by letting ( u = \ln(x) ) and ( dv = dx ). Consequently, we compute the derivatives:

• The derivative of ( u ) is given by ( \frac{du}{dx} = \frac{1}{x} ), which implies that ( du = \frac{1}{x} , dx ).
• The integral of ( dv ) gives us ( v = x ).

Using the integration by parts formula, which states that

we can substitute our values:

This simplifies to:

The integral of ( dx ) is simply ( x ), leading us to:

Thus, we conclude that the integral of ( \ln(x) , dx ) is

where $C$ is the constant of integration.

To verify our result, we can differentiate the expression ( x \ln(x) - x + C ) with respect to ( x ). The derivative is:

This confirms that the derivative of our result is indeed ( \ln(x) ), validating the correctness of our integral.