Evaluate. (Be sure to check by differentiating!)
\[
\int \frac{(\ln x)^{30}}{x} d x, x> 0
\]
\[
\int \frac{(\ln x)^{30}}{x} d x=
\]
(Type an exact answer. Use parentheses to clearly denote the argument of each function.)
Answer
Final Answer: \(\boxed{\frac{(\ln x)^{31}}{31} + C}\) where C is the constant of integration.
Step 1 ：The integral is of the form ∫f'(x)f(x) dx, where f(x) = (ln x)^30 and f'(x) = 1/x. This is a standard form of integral that can be solved by the method of integration by substitution.
Step 2 ：We can set u = f(x) = (ln x)^30, then du = f'(x) dx = 1/x dx. Substituting these into the integral, we get ∫u du, which is a simple integral to solve.
Step 3 ：After solving, we substitute back u = (ln x)^30 to get the final answer.
Step 4 ：Final Answer: \(\boxed{\frac{(\ln x)^{31}}{31} + C}\) where C is the constant of integration.
