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To solve the integral

\[
\int \sin^3(x) \cos^5(x) \, dx,
\]

we can use a substitution technique. Here’s the step-by-step solution:

### Step 1: Rewrite \(\sin^3(x)\)
Rewrite \(\sin^3(x)\) as \(\sin^2(x) \cdot \sin(x)\):
\[
\int \sin^3(x) \cos^5(x) \, dx = \int \sin^2(x) \cos^5(x) \sin(x) \, dx.
\]

### Step 2: Use a Pythagorean Identity
Since \(\sin^2(x) = 1 - \cos^2(x)\), substitute \(\sin^2(x)\) as follows:
\[
= \int (1 - \cos^2(x)) \cos^5(x) \sin(x) \, dx.
\]

### Step 3: Substitute \(u = \cos(x)\)
Let \(u = \cos(x)\). Then, \(du = -\sin(x) \, dx\), or equivalently, \(-du = \sin(x) \, dx\).

Substitute \(u\) and \(-du\) into the integral:
\[
= \int (1 - u^2) u^5 \cdot (-du).
\]

This simplifies to:
\[
= \int (u^5 - u^7) \, (-du).
\]

### Step 4: Distribute the Negative Sign
Rewrite as:
\[
= \int (-u^5 + u^7) \, du.
\]

### Step 5: Integrate Term by Term
Now integrate each term with respect to \(u\):
\[
= -\frac{u^6}{6} + \frac{u^8}{8} + C.
\]

### Step 6: Substitute Back \(u = \cos(x)\)
Replace \(u\) with \(\cos(x)\):
\[
= -\frac{\cos^6(x)}{6} + \frac{\cos^8(x)}{8} + C.
\]

### Final Answer
\[
\int \sin^3(x) \cos^5(x) \, dx = -\frac{\cos^6(x)}{6} + \frac{\cos^8(x)}{8} + C.
\]

Would you like further details, or do you have any questions?

### Related Questions
1. How would you integrate \(\int \sin^5(x) \cos^3(x) \, dx\) using a similar technique?
2. What happens if both sine and cosine powers are even, e.g., \(\int \sin^4(x) \cos^4(x) \, dx\)?
3. Can we solve \(\int \sin^3(x) \cos^3(x) \, dx\) using trigonometric identities?
4. How can we solve integrals involving higher powers, like \(\int \sin^7(x) \cos^9(x) \, dx\)?
5. What is the general form for integrating \(\int \sin^m(x) \cos^n(x) \, dx\) for arbitrary \(m\) and \(n\)?

### Tip
For integrals involving products of \(\sin\) and \(\cos\) raised to odd powers, using trigonometric identities and substitution is often the most efficient method.

Explore the step-by-step solution to the integral ∫ sin^3(x) cos^5(x) dx. This calculus problem involves trigonometric identities and substitution techniques, suitable for undergraduate students studying integral calculus.

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