To evaluate the integral \( \int 4 \sin^3(x) \cos^2(x) \, dx \), we can use trigonometric identities and substitution to simplify the expression. Here's the step-by-step solution:

### Step 1: Use a Trigonometric Identity
We know the identity for powers of sine and cosine, which is often helpful in integrals involving both:

\[
\sin^3(x) = \sin(x)(1 - \cos^2(x))
\]

Thus, we can rewrite the integral as:

\[
\int 4 \sin^3(x) \cos^2(x) \, dx = \int 4 \sin(x) (1 - \cos^2(x)) \cos^2(x) \, dx
\]

### Step 2: Expand the Integral
Now distribute \( \sin(x) \) to simplify the integrand:

\[
\int 4 \sin(x) ( \cos^2(x) - \cos^4(x)) \, dx = 4 \int \sin(x) \cos^2(x) \, dx - 4 \int \sin(x) \cos^4(x) \, dx
\]

### Step 3: Use Substitution
For both integrals, let \( u = \cos(x) \), which means \( du = -\sin(x) \, dx \). So the integrals become:

- First integral:
\[
4 \int \sin(x) \cos^2(x) \, dx = 4 \int u^2 (-du) = -4 \int u^2 \, du = -\frac{4}{3} u^3 = -\frac{4}{3} \cos^3(x)
\]

- Second integral:
\[
-4 \int \sin(x) \cos^4(x) \, dx = -4 \int u^4 (-du) = 4 \int u^4 \, du = \frac{4}{5} u^5 = \frac{4}{5} \cos^5(x)
\]

### Step 4: Combine the Results
Putting everything together, the result of the integral is:

\[
\int 4 \sin^3(x) \cos^2(x) \, dx = -\frac{4}{3} \cos^3(x) + \frac{4}{5} \cos^5(x) + C
\]

where \( C \) is the constant of integration.

### Final Answer:
\[
\boxed{-\frac{4}{3} \cos^3(x) + \frac{4}{5} \cos^5(x) + C}
\]

Would you like more details on any specific part? Also, here are some related questions to explore:

1. How would you evaluate \( \int \sin^2(x) \cos^2(x) \, dx \)?
2. Can you solve \( \int \sin^4(x) \cos^2(x) \, dx \)?
3. What happens if you use a different substitution, such as \( u = \sin(x) \), in this problem?
4. How can the method of trigonometric identities help in simplifying other integrals?
5. How would the integral change if it were \( \int \sin^4(x) \cos^3(x) \, dx \)?

**Tip:** Always try to use trigonometric identities when you have powers of sine and cosine in integrals, as they help to simplify the expressions.

Learn how to evaluate the integral of 4 sin^3(x) cos^2(x) step-by-step using trigonometric identities and substitution. This detailed solution is ideal for students in Grades 11-12 or early college, covering key concepts such as trigonometric integrals and substitution methods.

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