Evaluate. (Assume $x>0$.) Check by differentiating. \[ \int x^{8} \ln x d x \]

Step 1 ：Given the integral \(\int x^{8} \ln x dx\), we can use the method of integration by parts to solve it. The formula for integration by parts is \(\int u dv = uv - \int v du\), where u and v are functions of x.

Step 2 ：We choose \(u = \ln(x)\) and \(dv = x^8 dx\) because the derivative of \(\ln(x)\) is simpler than \(\ln(x)\) itself, and the antiderivative of \(x^8 dx\) is straightforward to find.

Step 3 ：We find that \(du = 1/x\) and \(v = x^9/9\).

Step 4 ：We then calculate \(uv = x^9\ln(x)/9\) and \(\int v du = x^9/81\).

Step 5 ：Substituting these values into the integration by parts formula, we find that the integral of the function \(x^8 \ln(x) dx\) is given by the expression \(x^9 \ln(x) / 9 - x^9 / 81\).

Step 6 ：Thus, the final answer is \(\boxed{\frac{x^{9} \ln x}{9} - \frac{x^{9}}{81}}\).