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To evaluate the integral 

\[
\int \sin^6(x) \cos^3(x) \, dx,
\]

we can use a substitution approach. Since we have a power of \(\cos(x)\), we can use the identity:

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

to express \(\sin^6(x)\) in terms of \(\cos(x)\), or alternatively use a substitution if we simplify with trigonometric identities.

Here’s a step-by-step approach:

### Step 1: Rewrite the Integral
Since \(\cos^3(x)\) is present, we can let \(u = \sin(x)\). Then, \(du = \cos(x) \, dx\), and \(\cos(x) \, dx = du\).

Notice that we have \(\cos^3(x) \, dx\) in the integral, which can be split as:
\[
\cos^2(x) \cos(x) \, dx = (1 - \sin^2(x)) \cos(x) \, dx.
\]

Letting \(u = \sin(x)\), then \(\sin^6(x) = u^6\), \(\cos(x) \, dx = du\), and \(\cos^2(x) = 1 - u^2\).

### Step 2: Substitute and Simplify
The integral now becomes:
\[
\int u^6 (1 - u^2) u \, du.
\]
Expanding this gives:
\[
\int (u^6 - u^8) u \, du = \int u^9 - u^{11} \, du.
\]

### Step 3: Integrate Term by Term
Now, integrate each term:
\[
\int u^9 \, du - \int u^{11} \, du = \frac{u^{10}}{10} - \frac{u^{12}}{12} + C.
\]

### Step 4: Substitute Back
Since \(u = \sin(x)\), we substitute back:
\[
\frac{\sin^{10}(x)}{10} - \frac{\sin^{12}(x)}{12} + C.
\]

### Final Answer:
\[
\int \sin^6(x) \cos^3(x) \, dx = \frac{\sin^{10}(x)}{10} - \frac{\sin^{12}(x)}{12} + C.
\]

Would you like more details on any specific step or have any questions?

Here are some related questions to explore further:

1. How would you approach this integral if the powers of \(\sin(x)\) and \(\cos(x)\) were different?
2. What other trigonometric identities could simplify integrals involving powers of \(\sin(x)\) and \(\cos(x)\)?
3. How does substitution work for integrals involving trigonometric functions?
4. Could integration by parts be useful in similar trigonometric integrals?
5. How would the approach change if the integral involved \(\tan(x)\) and \(\sec(x)\)?

**Tip:** When working with trigonometric integrals, always consider using identities or substitutions that reduce the powers of the trigonometric functions. This often simplifies the integration process.

Integrate \( \int \sin^6(x) \cos^3(x) \, dx \).

Explore the solution to the integral \( \int \sin^6(x) \cos^3(x) \, dx \) using trigonometric identities and substitution methods. This problem demonstrates power reduction and integration techniques for trigonometric functions, suitable for advanced high school math students (Grades 11-12).

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