Evaluate. (Assume $x> 0$.) Check by differentiating.
\[
\int x^{12} \ln x d x
\]
\[
\int x^{12} \ln x d x=
\]

Step 1 ：Identify the parts of the integral that will be 'u' and 'dv'. In this case, let 'u' be \(\ln x\) and 'dv' be \(x^{12} dx\).

Step 2 ：Find 'du' and 'v'. 'du' is the derivative of 'u' and 'v' is the integral of 'dv'. So, 'du' is \(\frac{1}{x}\) and 'v' is \(\frac{x^{13}}{13}\).

Step 3 ：Apply the formula for integration by parts, \(\int u dv = uv - \int v du\). This gives us \(\frac{x^{13}\ln x}{13} - \frac{x^{13}}{169}\).

Step 4 ：Check the result by differentiating it and see if we get back the original integrand. The derivative of \(\frac{x^{13}\ln x}{13} - \frac{x^{13}}{169}\) is \(x^{12}\ln x\), which is the original integrand.

Step 5 ：Final Answer: The integral of \(x^{12} \ln x dx\) is \(\boxed{\frac{x^{13}\ln x}{13} - \frac{x^{13}}{169}}\).