To find the indefinite integral of \( \sin^2(x) \cos^2(x) \, dx \), we can use a trigonometric identity to simplify the expression before integrating.

### Step 1: Use the double-angle identity
We can use the double-angle identity for sine:
\[
\sin(2x) = 2\sin(x)\cos(x)
\]
Squaring both sides:
\[
\sin^2(2x) = 4\sin^2(x)\cos^2(x)
\]
So,
\[
\sin^2(x)\cos^2(x) = \frac{1}{4} \sin^2(2x)
\]

### Step 2: Integrate using the identity
Now the integral becomes:
\[
\int \sin^2(x) \cos^2(x) \, dx = \frac{1}{4} \int \sin^2(2x) \, dx
\]

### Step 3: Use the power-reduction identity
To integrate \( \sin^2(2x) \), use the power-reduction identity:
\[
\sin^2(2x) = \frac{1 - \cos(4x)}{2}
\]
So the integral becomes:
\[
\frac{1}{4} \int \frac{1 - \cos(4x)}{2} \, dx = \frac{1}{8} \int (1 - \cos(4x)) \, dx
\]

### Step 4: Integrate term by term
Now, integrate each term separately:
\[
\frac{1}{8} \int 1 \, dx - \frac{1}{8} \int \cos(4x) \, dx
\]
\[
= \frac{1}{8} x - \frac{1}{8} \cdot \frac{\sin(4x)}{4} + C
\]
\[
= \frac{x}{8} - \frac{\sin(4x)}{32} + C
\]

### Final Answer
\[
\boxed{\int \sin^2(x) \cos^2(x) \, dx = \frac{x}{8} - \frac{\sin(4x)}{32} + C}
\]

Would you like to delve deeper into any step or have any questions?

Here are some related questions for further practice:
1. How would you integrate \( \sin^4(x) \, dx \)?
2. What is the indefinite integral of \( \cos^4(x) \, dx \)?
3. Can you find the indefinite integral of \( \cos^2(x) \, dx \)?
4. How do you solve \( \int \sin^2(2x) \, dx \)?
5. How would you integrate \( \sin(2x)\cos(2x) \, dx \)?

**Tip:** Whenever you encounter products of sine and cosine functions raised to powers, trigonometric identities like double-angle and power-reduction formulas can greatly simplify the integration process.

Learn how to integrate sin^2(x) cos^2(x) dx with a detailed step-by-step solution using trigonometric identities and integration techniques. This problem covers advanced integration concepts, including double-angle and power-reduction identities. Suitable for advanced high school and undergraduate students.

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