To integrate $\sin(x)(\cos(x))^2$, we proceed as follows:

Given:

$\int \sin(x)(\cos(x))^2 \, dx$

Step-by-Step Solution:

1. Substitute: Let $u = \cos(x)$.
   Then, $du = -\sin(x) \, dx$.

   The integral becomes: $\int \sin(x)(\cos(x))^2 \, dx = -\int u^2 \, du$

2. Integrate: $-\int u^2 \, du = -\frac{u^3}{3} + C$

3. Back-substitute: Replace $u$ with $\cos(x)$: $-\frac{u^3}{3} + C = -\frac{(\cos(x))^3}{3} + C$

Final Answer:

$\int \sin(x)(\cos(x))^2 \, dx = -\frac{(\cos(x))^3}{3} + C$

Would you like additional clarification on the process?

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Related Questions:

1. How to handle integrals involving $(\sin(x))^n (\cos(x))^m$?
2. What is the difference between substitution and integration by parts?
3. Can we solve this integral without substitution?
4. What is the geometric interpretation of this integral?
5. How does this technique generalize to other trigonometric integrals?

Tip:

Always look for substitutions that simplify the integral into a basic polynomial or known function.