To solve this problem, let's go through each step of integration by parts.

Given Integral

We need to evaluate: $\int \frac{\ln(x)}{x^5} \, dx$

Step 1: Choosing $f(x)$ and $g'(x)$

For integration by parts, we need to choose parts $f(x)$ and $g'(x)$ such that: $\int f(x) g'(x) \, dx = f(x) g(x) - \int f'(x) g(x) \, dx$

In this case, a good choice is:

• $f(x) = \ln(x)$ (since the derivative of $\ln(x)$ is simpler, and it will reduce the complexity of the integral)
• $g'(x) = \frac{1}{x^5}$

Step 2: Finding $f'(x)$ and $g(x)$

1. Differentiate $f(x) = \ln(x)$: $f'(x) = \frac{1}{x}$

2. Integrate $g'(x) = \frac{1}{x^5}$: $g(x) = \int \frac{1}{x^5} \, dx = \int x^{-5} \, dx = \frac{x^{-4}}{-4} = -\frac{1}{4x^4}$

So we have:

• $f(x) = \ln(x)$
• $g'(x) = \frac{1}{x^5}$
• $f'(x) = \frac{1}{x}$
• $g(x) = -\frac{1}{4x^4}$

Step 3: Applying Integration by Parts

Now, using the integration by parts formula: $\int \frac{\ln(x)}{x^5} \, dx = f(x) g(x) - \int f'(x) g(x) \, dx$ Substitute the values we found: $= \ln(x) \cdot \left(-\frac{1}{4x^4}\right) - \int \frac{1}{x} \cdot \left(-\frac{1}{4x^4}\right) \, dx$ $= -\frac{\ln(x)}{4x^4} + \int \frac{1}{4x^5} \, dx$

Step 4: Simplifying the Integral

Now, integrate $\frac{1}{4x^5}$: $\int \frac{1}{4x^5} \, dx = \frac{1}{4} \int x^{-5} \, dx = \frac{1}{4} \cdot \frac{x^{-4}}{-4} = -\frac{1}{16x^4}$

Final Answer

Combine the terms: $\int \frac{\ln(x)}{x^5} \, dx = -\frac{\ln(x)}{4x^4} - \frac{1}{16x^4} + C$

Summary

The evaluated integral is: $\int \frac{\ln(x)}{x^5} \, dx = -\frac{\ln(x)}{4x^4} - \frac{1}{16x^4} + C$

Would you like a detailed explanation of any step or have further questions?

Relative Questions

1. How do you determine the best choice for $f(x)$ and $g'(x)$ in integration by parts?
2. Why is integration by parts useful for integrals involving logarithmic functions?
3. What would change if the integral were $\int \frac{\ln(x)}{x^4} \, dx$ instead?
4. Can we use substitution in conjunction with integration by parts, and if so, how?
5. How does the choice of $f(x)$ and $g(x)$ affect the complexity of the solution?

Tip

When choosing parts for integration by parts, it’s generally helpful to pick $f(x)$ as a function that becomes simpler when differentiated (like logarithmic or polynomial functions).