Determine \( \int y d x \) in each of the following cases: \( y=\frac{\ln x}{x} \)

To solve \( \int y \, dx \) for \( y = \frac{\ln x}{x} \), we first rewrite the integral: \[ \int \frac{\ln x}{x} \, dx \] This integral requires a substitution. Let \( u = \ln x \), which means that \( du = \frac{1}{x} \, dx \). Therefore, we have: \[ dx = x \, du = e^u \, du \] Now, substituting \( x = e^u \) into our integral gives us: \[ \int u \, du \] This integral is straightforward: \[ \int u \, du = \frac{u^2}{2} + C = \frac{(\ln x)^2}{2} + C \] Thus, the solution to the original integral is: \[ \int \frac{\ln x}{x} \, dx = \frac{(\ln x)^2}{2} + C \] where \( C \) is the constant of integration.