We need to evaluate the integral:

$\int x e^{8x} \, dx$

This requires integration by parts, which uses the formula:

$\int u \, dv = uv - \int v \, du$

Step 1: Assign $u$ and $dv$

Let: $u = x \quad \text{and} \quad dv = e^{8x} dx$

Now, compute $du$ and $v$: $du = dx \quad \text{and} \quad v = \frac{1}{8} e^{8x}$

Step 2: Apply the Integration by Parts Formula

Substitute into the formula: $\int x e^{8x} dx = uv - \int v \, du$

$\int x e^{8x} dx = x \cdot \frac{1}{8} e^{8x} - \int \frac{1}{8} e^{8x} dx$

Step 3: Simplify the Expression

The first term is: $\frac{x}{8} e^{8x}$

The remaining integral is: $\int \frac{1}{8} e^{8x} dx = \frac{1}{8} \cdot \frac{1}{8} e^{8x} = \frac{1}{64} e^{8x}$

Thus, the solution becomes: $\int x e^{8x} dx = \frac{x}{8} e^{8x} - \frac{1}{64} e^{8x} + C$

Final Answer:

$\int x e^{8x} dx = \frac{x}{8} e^{8x} - \frac{1}{64} e^{8x} + C$

Would you like a detailed breakdown of any specific step or have any questions?

Related Questions:

1. How is the formula for integration by parts derived?
2. Can this integral be solved without using integration by parts?
3. What is the importance of the constant of integration?
4. How does integration by parts differ from substitution methods?
5. Can integration by parts be applied recursively? What are some examples?

Tip:

When choosing $u$ and $dv$ for integration by parts, prioritize selecting $u$ as a term that simplifies upon differentiation!