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The integral you're asking for is:

\[
\int 8x \sin(x) \, dx
\]

To solve this, we will apply the technique of **integration by parts**. The integration by parts formula is:

\[
\int u \, dv = uv - \int v \, du
\]

### Step 1: Choose \( u \) and \( dv \)
Let:
- \( u = 8x \) (so that \( du = 8 \, dx \))
- \( dv = \sin(x) \, dx \) (so that \( v = -\cos(x) \))

### Step 2: Apply the formula
Using the formula \( \int u \, dv = uv - \int v \, du \), we get:

\[
\int 8x \sin(x) \, dx = 8x(-\cos(x)) - \int (-\cos(x)) \cdot 8 \, dx
\]

Simplifying:

\[
= -8x \cos(x) + 8 \int \cos(x) \, dx
\]

### Step 3: Solve the remaining integral
The integral of \( \cos(x) \) is \( \sin(x) \), so:

\[
8 \int \cos(x) \, dx = 8 \sin(x)
\]

### Step 4: Final result
Substituting back:

\[
\int 8x \sin(x) \, dx = -8x \cos(x) + 8 \sin(x) + C
\]

Where \( C \) is the constant of integration.

### Solution:
\[
\boxed{-8x \cos(x) + 8 \sin(x) + C}
\]

Would you like further details or have any questions?

Here are 5 follow-up questions you may consider:
1. How does the integration by parts method work in general?
2. What are other scenarios where integration by parts is useful?
3. How can you determine the best choice for \( u \) and \( dv \) when using integration by parts?
4. What happens if we choose different \( u \) and \( dv \) for this problem?
5. How do you handle integrals involving products of functions like \( x e^x \) or \( x \ln(x) \)?

**Tip:** For integrals involving products of polynomial and trigonometric (or exponential) functions, integration by parts is usually the best approach.

Learn how to compute the integral of 8x sin(x) using the integration by parts method. This detailed solution includes a step-by-step explanation, covering key concepts of calculus and integration techniques. Suitable for high school or university students.

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