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The integral provided is:

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

To solve this, we can use substitution and trigonometric identities.

### Step 1: Rewrite the integral

We know that \(\sin^3(x)\) can be rewritten as:

\[
\sin^3(x) = \sin(x) \cdot \sin^2(x)
\]

Using the identity \(\sin^2(x) = 1 - \cos^2(x)\), we get:

\[
\sin^3(x) = \sin(x) \cdot (1 - \cos^2(x))
\]

So, the integral becomes:

\[
\int \cos(x) \cdot \sin(x) \cdot (1 - \cos^2(x)) \, dx
\]

### Step 2: Substitution

Let \( u = \sin(x) \), hence \( du = \cos(x) \, dx \). The integral simplifies to:

\[
\int u \cdot (1 - u^2) \, du
\]

### Step 3: Simplify and Integrate

Expanding the integrand:

\[
\int u \cdot (1 - u^2) \, du = \int (u - u^3) \, du
\]

Now, integrate term by term:

\[
\int u \, du - \int u^3 \, du = \frac{u^2}{2} - \frac{u^4}{4} + C
\]

### Step 4: Back Substitution

Since \( u = \sin(x) \), substitute back:

\[
\frac{\sin^2(x)}{2} - \frac{\sin^4(x)}{4} + C
\]

Thus, the solution to the integral is:

\[
\boxed{\frac{\sin^2(x)}{2} - \frac{\sin^4(x)}{4} + C}
\]

Would you like a detailed explanation of any particular step? Here are a few related questions to practice:

1. How would you solve \(\int \sin^3(x) \, dx\)?
2. What substitution method can be used for integrals involving products of sine and cosine?
3. Can you express \(\cos^3(x)\) using trigonometric identities?
4. How does integration by parts apply to integrals like this?
5. How do you handle definite integrals of trigonometric functions?

**Tip:** When solving integrals involving powers of sine or cosine, using trigonometric identities often simplifies the problem.

Find the indefinite integral of cos(x) sin^3(x) dx.

Learn how to find the indefinite integral of cos(x) sin^3(x) dx using substitution and trigonometric identities. This detailed step-by-step solution covers key concepts in integration and trigonometry, including the substitution method. Suitable for advanced high school and college-level students.

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