In the context of the given integral \(\int \cos^3(x) \sin(x) \, dx\), the issue of \(\sin(x)\) being negative or positive does not affect the method of integration. What we are focused on is using substitution to simplify the integral. 

The substitution \( u = \cos(x) \) and the differential relationship \( du = -\sin(x) \, dx \) are mathematically consistent regardless of the sign of \(\sin(x)\). Here's a step-by-step breakdown to ensure clarity:

1. **Substitution**: We start with \( u = \cos(x) \). Consequently,
   \[
   \frac{du}{dx} = -\sin(x) \Rightarrow du = -\sin(x) \, dx.
   \]
   Here, \( \sin(x) \, dx = -du \). This relationship holds true for any value of \(\sin(x)\), whether positive or negative.

2. **Integral Transformation**: Substituting \( u = \cos(x) \) and \( \sin(x) \, dx = -du \) into the integral,
   \[
   \int \cos^3(x) \sin(x) \, dx = \int u^3 (-du) = -\int u^3 \, du.
   \]

3. **Integration**: We then integrate \( u^3 \),
   \[
   -\int u^3 \, du = -\frac{u^4}{4} + C.
   \]

4. **Back Substitution**: Finally, substitute back \( u = \cos(x) \),
   \[
   -\frac{u^4}{4} + C = -\frac{\cos^4(x)}{4} + C.
   \]

Thus, the final result is:
\[
\boxed{-\frac{\cos^4(x)}{4} + C}.
\]

The critical step is the substitution, which is independent of whether \(\sin(x)\) is positive or negative. The sign of \(\sin(x)\) merely reflects in the differential \( du = -\sin(x) \, dx \), ensuring the consistency and correctness of the integration process.

Would you like further details or have any questions? Here are some related questions for further practice:

1. \(\int \cos^2(x) \sin(x) \, dx\)
2. \(\int \cos^3(x) \, dx\)
3. \(\int \sin^3(x) \cos(x) \, dx\)
4. \(\int \cos^5(x) \sin(x) \, dx\)
5. \(\int \sin^4(x) \cos(x) \, dx\)
6. \(\int \cos^2(x) \, dx\)
7. \(\int \cos(x) \, dx\)
8. \(\int \sin(x) \, dx\)

**Tip:** Consistent substitution is a powerful method in integration. Make sure to transform all parts of the integral, including \( dx \), according to the substitution.

Explore the concept of integration involving trigonometric substitution and understand why the sign of sin(x) does not affect the integration process. Step-by-step solution provided with examples. Suitable for advanced mathematics enthusiasts.

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