To find the integral of \( \ln(\sqrt{x}) \), we can rewrite it in a more convenient form.

First, recall that \( \sqrt{x} = x^{1/2} \), so:

\[
\ln(\sqrt{x}) = \ln(x^{1/2}) = \frac{1}{2} \ln(x)
\]

Now, the problem reduces to integrating \( \frac{1}{2} \ln(x) \):

\[
\int \ln(\sqrt{x}) \, dx = \frac{1}{2} \int \ln(x) \, dx
\]

We know the standard integral of \( \ln(x) \) is:

\[
\int \ln(x) \, dx = x \ln(x) - x + C
\]

Thus, the integral becomes:

\[
\frac{1}{2} \int \ln(x) \, dx = \frac{1}{2} \left( x \ln(x) - x \right) + C
\]

Simplifying this expression:

\[
\int \ln(\sqrt{x}) \, dx = \frac{1}{2} x \ln(x) - \frac{1}{2} x + C
\]

### Final Answer:
\[
\int \ln(\sqrt{x}) \, dx = \frac{1}{2} x \ln(x) - \frac{1}{2} x + C
\]

Would you like any further details or clarifications?

### 5 Related Questions:
1. How do you integrate more complex logarithmic functions?
2. What is the integral of \( \ln(x^n) \) for any positive integer \( n \)?
3. Can the integration of other root functions with logarithms be done similarly?
4. How do you solve integrals involving products of \( \ln(x) \) and polynomials?
5. What techniques are used to evaluate integrals with logarithmic functions in more advanced calculus?

### Tip:
When dealing with logarithmic functions in integrals, remember to simplify expressions using logarithmic properties like \( \ln(a^b) = b\ln(a) \), as it often makes integration easier.

Find the integral of ln(sqrt(x)) with a detailed step-by-step solution, using logarithmic properties and standard integrals. This calculus problem is suitable for high school students in Grades 11-12 studying integration techniques.

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