To solve \( \int \ln(\sqrt{x}) \, dx \) using **integration by parts**, let's first set it up.

### Step 1: Rewriting the integral
As before, rewrite \( \ln(\sqrt{x}) \) as:

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

Thus, the integral becomes:

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

Now, focus on solving \( \int \ln(x) \, dx \) using **integration by parts**.

### Step 2: Set up integration by parts
We use the formula for integration by parts:

\[
\int u \, dv = uv - \int v \, du
\]

Let:

- \( u = \ln(x) \), so \( du = \frac{1}{x} \, dx \)
- \( dv = dx \), so \( v = x \)

### Step 3: Apply the formula
Now, apply the integration by parts formula:

\[
\int \ln(x) \, dx = x \ln(x) - \int x \cdot \frac{1}{x} \, dx
\]

Simplifying:

\[
\int \ln(x) \, dx = x \ln(x) - \int 1 \, dx
\]

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

### Step 4: Go back to the original integral
Now, substitute this result back into the original integral:

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

### Step 5: Simplify the expression
\[
\int \ln(\sqrt{x}) \, dx = \frac{1}{2} x \ln(x) - \frac{1}{2} x + C
\]

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

This is the same result as before, now derived using **integration by parts**.

Would you like more clarification on this method?

### 5 Related Questions:
1. How do you choose \( u \) and \( dv \) in integration by parts

Learn how to solve the integral of ln(√x) using integration by parts, with a detailed step-by-step solution. This problem involves calculus concepts like logarithmic properties and the integration by parts technique. Suitable for advanced high school and college-level students.

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