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

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Calculate the integral $$ \int \frac{\ln x}{x} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{\ln{\left(xight)}}{x} dx$$
- step1: Use the substitution $$dx=x dt$$ to transform the integral$$:$$
  $$\int \frac{\ln{\left(xight)}}{x}	imes x dt$$
- step2: Simplify:
  $$\int \ln{\left(xight)} dt$$
- step3: Use the substitution $$t=\ln{\left(xight)}$$ to transform the integral$$:$$
  $$\int t dt$$
- step4: Evaluate the integral:
  $$\frac{t^{1+1}}{1+1}$$
- step5: Add the numbers:
  $$\frac{t^{2}}{1+1}$$
- step6: Add the numbers:
  $$\frac{t^{2}}{2}$$
- step7: Substitute back:
  $$\frac{\left(\ln{\left(xight)}ight)^{2}}{2}$$
- step8: Simplify the expression:
  $$\frac{1}{2}\left(\ln{\left(xight)}ight)^{2}$$
- step9: Add the constant of integral C:
  $$\frac{1}{2}\left(\ln{\left(xight)}ight)^{2} + C, C \in \mathbb{R}$$
The integral of $$ y = \frac{\ln x}{x} $$ with respect to $$ x $$ is $$ \frac{1}{2}(\ln x)^2 + C $$, where $$ C $$ is an arbitrary constant.
The integral of $$ \frac{\ln x}{x} $$ with respect to $$ x $$ is $$ \frac{1}{2}(\ln x)^2 + C $$.

Determine in each of the following cases:

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