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To solve the given indefinite integral:

\[
\int \sin^3(x) \cos^2(x) \, dx
\]

we can proceed by simplifying the expression using trigonometric identities.

### Step 1: Rewrite \(\sin^3(x)\)
We know that:
\[
\sin^3(x) = \sin(x) \cdot \sin^2(x)
\]
and using the identity \(\sin^2(x) = 1 - \cos^2(x)\), we have:
\[
\sin^3(x) = \sin(x)(1 - \cos^2(x))
\]

So the integral becomes:
\[
\int \sin(x)(1 - \cos^2(x)) \cos^2(x) \, dx
\]

### Step 2: Use substitution
Let \( u = \cos(x) \), which implies \( du = -\sin(x) \, dx \).

Thus, the integral transforms into:
\[
-\int (1 - u^2) u^2 \, du
\]
Expanding the integrand:
\[
-\int (u^2 - u^4) \, du
\]

### Step 3: Integrate
Now, integrate term by term:
\[
-\left( \frac{u^3}{3} - \frac{u^5}{5} \right) + C
\]
where \( C \) is the constant of integration.

### Step 4: Substitute back \( u = \cos(x) \)
Finally, substitute back \( u = \cos(x) \) to get the result in terms of \( x \):
\[
-\left( \frac{\cos^3(x)}{3} - \frac{\cos^5(x)}{5} \right) + C
\]
This is the indefinite integral.

Would you like further clarification or details about any step? 

Here are 5 related questions that can help deepen understanding:

1. How does the substitution method work in integration, and why is it useful?
2. Can we apply other trigonometric identities to simplify different integrals?
3. How would the process differ if we were given \(\sin^2(x)\cos^3(x)\)?
4. What happens if we are asked to compute the definite integral over a specific interval?
5. Could we solve this using integration by parts instead of substitution?

**Tip:** Always consider using trigonometric identities to simplify integrals involving powers of sine and cosine.

Find the indefinite integral: ∫ sin^3(x) cos^2(x) dx

Learn how to solve the indefinite integral ∫ sin^3(x) cos^2(x) dx using the substitution method. The solution involves breaking down trigonometric functions and applying integration techniques. Suitable for students studying college-level calculus.

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