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.