Find the indefinite integral. (Note: Solve by the simplest method-not all require integration by parts. Remember the constant of integration.) \[ \int \frac{(\ln (x))^{9}}{x} d x \]

To solve the integral \(\int \frac{(\ln (x))^{9}}{x} \, dx\), we can use a straightforward substitution. Let’s set \(u = \ln(x)\). Then, the derivative of \(u\) is \(du = \frac{1}{x} \, dx\), which means that \(dx = x \, du = e^{u} \, du\). Rewriting the integral in terms of \(u\), we have: \[ \int \frac{(u)^{9}}{e^{u}} \cdot e^{u} \, du = \int u^9 \, du \] Now, integrating \(u^{9}\) gives us: \[ \frac{u^{10}}{10} + C \] Substituting back \(u = \ln(x)\), we have: \[ \frac{(\ln(x))^{10}}{10} + C \] Thus, the indefinite integral is: \[ \int \frac{(\ln (x))^{9}}{x} \, dx = \frac{(\ln(x))^{10}}{10} + C \]