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The given integral is:

\[
\int \cos^3(x) \cdot \sin(x) \, dx
\]

To solve this, we can use a substitution method. Let's break it down step by step.

### Step 1: Use substitution
We will use the substitution \( u = \cos(x) \). Therefore:

\[
du = -\sin(x) \, dx
\]

So the given integral becomes:

\[
-\int u^3 \, du
\]

### Step 2: Integrate
Now we integrate:

\[
-\int u^3 \, du = -\frac{u^4}{4} + C
\]

### Step 3: Substitute back \( u = \cos(x) \)
Substituting \( u = \cos(x) \) back into the equation, we get:

\[
-\frac{\cos^4(x)}{4} + C
\]

Thus, the solution to the integral is:

\[
-\frac{\cos^4(x)}{4} + C
\]

Would you like further details or have any questions? Here are 5 related questions to deepen your understanding:

1. How does the substitution method help simplify integrals involving trigonometric functions?
2. What other substitution could work for similar integrals with sine and cosine powers?
3. Can this integral be solved using integration by parts? Why or why not?
4. What is the general approach for integrating products of powers of sine and cosine?
5. How would this problem change if the exponent on \(\cos(x)\) was different?

**Tip:** When dealing with powers of trigonometric functions, substitution is often the most efficient method. Try using identities when substitution isn't clear right away!

Solve the integral of cos^3(x) * sin(x) using the substitution method. This problem involves basic trigonometric functions and the use of substitution for simplifying the integration process. Suitable for students in Grades 10-12.

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