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The problem asks to use integration by parts to solve the integral:

\[
\int (\ln x)^2 \, dx
\]

We can solve this using the method of integration by parts, which is given by:

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

### Step-by-step Solution:

1. **Choose \( u \) and \( dv \):**

   We set:
   \[
   u = (\ln x)^2 \quad \text{so that} \quad du = 2 \ln x \cdot \frac{1}{x} \, dx
   \]
   And,
   \[
   dv = dx \quad \text{so that} \quad v = x
   \]

2. **Apply the Integration by Parts Formula:**

   Using the formula \( \int u \, dv = uv - \int v \, du \), we have:
   \[
   \int (\ln x)^2 \, dx = x (\ln x)^2 - \int x \cdot 2 \ln x \cdot \frac{1}{x} \, dx
   \]
   Simplify the second term:
   \[
   \int x \cdot 2 \ln x \cdot \frac{1}{x} \, dx = \int 2 \ln x \, dx
   \]

3. **Solve \( \int 2 \ln x \, dx \):**

   Again, we apply integration by parts for \( \int 2 \ln x \, dx \).

   Let:
   \[
   u = \ln x \quad \text{so that} \quad du = \frac{1}{x} \, dx
   \]
   And,
   \[
   dv = dx \quad \text{so that} \quad v = x
   \]

   Now apply the formula:
   \[
   \int 2 \ln x \, dx = 2 (x \ln x - \int x \cdot \frac{1}{x} \, dx)
   \]
   Simplify:
   \[
   \int 2 \ln x \, dx = 2 (x \ln x - x) = 2x \ln x - 2x
   \]

4. **Substitute Back:**

   Now substitute back into the original expression:
   \[
   \int (\ln x)^2 \, dx = x (\ln x)^2 - (2x \ln x - 2x) + C
   \]
   Simplify:
   \[
   \int (\ln x)^2 \, dx = x (\ln x)^2 - 2x \ln x + 2x + C
   \]

Thus, the integral is:

\[
\int (\ln x)^2 \, dx = x \left( (\ln x)^2 - 2 \ln x + 2 \right) + C
\]

Would you like further clarification or additional details on any steps?

---

Here are some related questions to explore:

1. How does integration by parts generally work?
2. Why was \( u = (\ln x)^2 \) chosen in this case?
3. What other methods can be used to integrate functions involving \( \ln x \)?
4. How can integration by parts be extended to higher powers of \( \ln x \)?
5. What are the applications of such logarithmic integrals in physics or engineering?

**Tip:** When applying integration by parts, always choose \( u \) and \( dv \) carefully. Selecting \( u \) as the part that simplifies upon differentiation often helps streamline the solution.

Use integration by parts to find ∫(ln x)^2 dx.

Learn how to use integration by parts to solve the integral ∫(ln x)^2 dx. This step-by-step solution covers key calculus concepts, such as integration by parts and logarithmic functions. Suitable for students at the undergraduate calculus level.

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