To find the indefinite integral of \( \sin^3(x) \cos^2(x) \, dx \), we can use trigonometric identities and substitution methods. Here’s a step-by-step solution:

### Step 1: Rewrite the integrand using trigonometric identities
We know the identity:
\[
\sin^2(x) = 1 - \cos^2(x)
\]
So we can rewrite \( \sin^3(x) \) as:
\[
\sin^3(x) = \sin(x) \cdot \sin^2(x) = \sin(x) \cdot (1 - \cos^2(x))
\]
Thus, the integral becomes:
\[
\int \sin^3(x) \cos^2(x) \, dx = \int \sin(x) (1 - \cos^2(x)) \cos^2(x) \, dx
\]
Expanding the integrand:
\[
\int \sin(x) \cos^2(x) \, dx - \int \sin(x) \cos^4(x) \, dx
\]

### Step 2: Use substitution
Let \( u = \cos(x) \), then \( du = -\sin(x) \, dx \). The integrals become:
\[
-\int u^2 \, du + \int u^4 \, du
\]
The negative sign comes from the substitution \( du = -\sin(x) \, dx \).

### Step 3: Integrate
Now integrate each term:
\[
-\int u^2 \, du = -\frac{u^3}{3} = -\frac{\cos^3(x)}{3}
\]
\[
\int u^4 \, du = \frac{u^5}{5} = \frac{\cos^5(x)}{5}
\]

### Step 4: Combine the results
Combine the results to get the final answer:
\[
\int \sin^3(x) \cos^2(x) \, dx = -\frac{\cos^3(x)}{3} + \frac{\cos^5(x)}{5} + C
\]
where \( C \) is the constant of integration.

### Final Answer
\[
\boxed{\int \sin^3(x) \cos^2(x) \, dx = -\frac{\cos^3(x)}{3} + \frac{\cos^5(x)}{5} + C}
\]

Would you like a detailed explanation of any particular step or have any questions?

Here are some related questions you might find interesting:
1. What is the indefinite integral of \( \sin^2(x) \cos^3(x) \, dx \)?
2. How would you integrate \( \sin^4(x) \cos^2(x) \, dx \)?
3. Can you find the indefinite integral of \( \sin^2(x) \cos^4(x) \, dx \)?
4. What substitution would you use for \( \sin^3(x) \cos(x) \, dx \)?
5. How would you solve \( \int \sin(x) \cos(x) \, dx \) using a double-angle identity?

**Tip:** When integrating trigonometric functions raised to powers, always consider using identities or substitutions to simplify the expression.

Explore how to integrate sin^3(x) cos^2(x) dx using trigonometric identities and substitution methods. Follow a detailed step-by-step solution to solve this advanced high school-level calculus problem.

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