$\int \frac{(\ln x)^{8}}{x} d x, x> 0$

Step 1 ：This is an integral problem. The integral is of the form \(\int \frac{(\ln x)^{n}}{x} dx, x>0\). This can be solved using the power rule for integration and the chain rule.

Step 2 ：The power rule states that the integral of \(x^n dx\) is \((1/n+1)x^(n+1) + C\), where C is the constant of integration.

Step 3 ：The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. In this case, the outer function is the natural logarithm function and the inner function is x.

Step 4 ：The derivative of the natural logarithm function is \(1/x\) and the derivative of x is 1. Therefore, the derivative of the composite function is \(1/x * 1 = 1/x\). This means that the integral of \(1/x dx\) is \(\ln|x| + C\).

Step 5 ：Therefore, the integral of \((\ln x)^8 / x dx\) is \((1/9)(\ln x)^9 + C\).

Step 6 ：The integral of the function \(\frac{(\ln x)^{8}}{x}\) with respect to x is \(\frac{(\ln x)^{9}}{9} + C\), where C is the constant of integration.

Step 7 ：Final Answer: \(\boxed{\frac{(\ln x)^{9}}{9} + C}\)