To integrate the function $\frac{\ln(x)}{x^8} \, dx$, we can use integration by parts. Let’s go through the steps:

Step 1: Set up Integration by Parts

For integration by parts, we use the formula: $\int u \, dv = u v - \int v \, du$ We need to choose $u$ and $dv$ carefully. A good choice here is:

• $u = \ln(x)$ (since the derivative of $\ln(x)$ simplifies nicely),
• $dv = \frac{1}{x^8} \, dx = x^{-8} \, dx$.

Step 2: Differentiate $u$ and Integrate $dv$

• Differentiate $u = \ln(x)$: $du = \frac{1}{x} \, dx$

• Integrate $dv = x^{-8} \, dx$: $v = \frac{x^{-7}}{-7} = -\frac{1}{7x^7}$

Step 3: Apply the Integration by Parts Formula

Now, apply the formula: $\int \frac{\ln(x)}{x^8} \, dx = u v - \int v \, du$ Substitute $u = \ln(x)$, $v = -\frac{1}{7x^7}$, and $du = \frac{1}{x} \, dx$: $= \ln(x) \cdot \left( -\frac{1}{7x^7} \right) - \int \left( -\frac{1}{7x^7} \right) \cdot \frac{1}{x} \, dx$ $= -\frac{\ln(x)}{7x^7} + \frac{1}{7} \int \frac{1}{x^8} \, dx$

Step 4: Integrate the Remaining Term

Now, integrate $\int \frac{1}{x^8} \, dx$: $\int x^{-8} \, dx = \frac{x^{-7}}{-7} = -\frac{1}{7x^7}$

Step 5: Combine the Results

Thus, the integral becomes: $\int \frac{\ln(x)}{x^8} \, dx = -\frac{\ln(x)}{7x^7} - \frac{1}{49x^7} + C$ where $C$ is the constant of integration.

Final Answer:

$\boxed{\int \frac{\ln(x)}{x^8} \, dx = -\frac{\ln(x)}{7x^7} - \frac{1}{49x^7} + C}$

Would you like more details on any of these steps, or do you have any questions?

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Here are 5 related questions:

1. How do you choose $u$ and $dv$ in integration by parts?
2. Can integration by parts be used for other logarithmic functions?
3. How do you handle improper integrals involving logarithms?
4. What are the general applications of integration by parts in physics?
5. Can you generalize the result for $\frac{\ln(x)}{x^n}$?

Tip: When performing integration by parts, always choose $u$ to be the part of the integrand that simplifies upon differentiation.