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.