To compute the integral of the function $e^x \sin(x) \ln(x)$, we will apply the method of integration by parts. We start by choosing $u = \ln(x)$ and $dv = e^x \sin(x) \, dx$.

Applying integration by parts, we have:

This simplifies to:

Next, we will apply integration by parts again to the integral $\int e^x \cos(x) \, dx$. This time, we set $u = \cos(x)$ and $dv = e^x \, dx$:

Substituting this back gives us:

Now, simplifying this expression results in:

Next, we will perform integration by parts one more time on the integral $\int e^x \sin(x) \, dx$, with $u = \sin(x)$ and $dv = e^x \, dx$:

Substituting this back into our earlier expression gives us:

After simplifying, we arrive at:

where $C$ is the constant of integration.

Thus, we conclude that the integral of $e^x \sin(x) \ln(x)$ is given by: