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The integral to solve is:

\[
\int 6x^2 \sin^3 x \, dx
\]

This integral involves a product of powers of \( x \) and trigonometric functions. We will approach this using integration by parts or reduction formulas, depending on the structure of the function.

### Solution Outline:

1. **Rewrite \( \sin^3 x \)** using trigonometric identities:
   \[
   \sin^3 x = \sin x (1 - \cos^2 x)
   \]
   Expanding this gives:
   \[
   \sin^3 x = \sin x - \sin x \cos^2 x
   \]
   
   So the integral becomes:
   \[
   \int 6x^2 (\sin x - \sin x \cos^2 x) \, dx
   \]
   Now break it into two integrals:
   \[
   \int 6x^2 \sin x \, dx - \int 6x^2 \sin x \cos^2 x \, dx
   \]

2. **Apply Integration by Parts**: We handle these terms using integration by parts and substitution, starting with the simpler integral \( \int 6x^2 \sin x \, dx \).

I will now work through the detailed solution of both integrals.The first part of the integral, \(\int 6x^2 \sin x \, dx\), results in:

\[
-6x^2 \cos x + 12x \sin x + 12 \cos x
\]

Now, we will solve the second part of the integral:

\[
\int 6x^2 \sin x \cos^2 x \, dx
\]

This will require additional techniques such as trigonometric identities or substitution. I will calculate this next.The second part of the integral, \(\int 6x^2 \sin x \cos^2 x \, dx\), results in:

\[
-2x^2 \cos^3 x + \frac{8x \sin^3 x}{3} + 4x \sin x \cos^2 x + \frac{8 \sin^2 x \cos x}{3} + \frac{28 \cos^3 x}{9}
\]

### Full Solution:

The original integral can now be written as the sum of the two parts:

\[
\int 6x^2 \sin^3 x \, dx = \left( -6x^2 \cos x + 12x \sin x + 12 \cos x \right) - \left( 2x^2 \cos^3 x - \frac{8x \sin^3 x}{3} - 4x \sin x \cos^2 x - \frac{8 \sin^2 x \cos x}{3} - \frac{28 \cos^3 x}{9} \right)
\]

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

### Related Questions:

1. How can substitution help in solving integrals involving trigonometric functions?
2. What are other useful trigonometric identities for integrals?
3. Can you apply integration by parts to trigonometric integrals systematically?
4. How do you approach integrals that involve products of powers and trigonometric functions?
5. What is the reduction formula for powers of sine or cosine?

### Tip:
When dealing with complex trigonometric integrals, always consider simplifying with identities or breaking the integral into smaller, manageable parts!

Learn how to solve the integral ∫ 6x^2 sin^3(x) dx by applying integration by parts and trigonometric identities. A step-by-step guide suitable for undergraduate calculus students.

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