To solve the definite integral

$\int_{1}^{3} \frac{8 (\ln x)^3}{x} \, dx,$

we follow these steps:

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Step 1: Recognize the Integration Formula

The integral is of the form: $\int \frac{(\ln x)^n}{x} \, dx = \frac{(\ln x)^{n+1}}{n+1} + C.$

Here, $n = 3$, so we use this formula in the definite integral.

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Step 2: Evaluate the Definite Integral

First, compute the indefinite integral: $\int \frac{8 (\ln x)^3}{x} \, dx = 8 \cdot \frac{(\ln x)^4}{4}.$

This simplifies to: $2 (\ln x)^4 + C.$

Now evaluate the definite integral from 1 to 3: $\int_{1}^{3} \frac{8 (\ln x)^3}{x} \, dx = \left[ 2 (\ln x)^4 \right]_1^3.$

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Step 3: Substitute the Limits

Substitute the upper limit $x = 3$: $2 (\ln 3)^4.$

Substitute the lower limit $x = 1$: $2 (\ln 1)^4.$

Since $\ln 1 = 0$, the lower limit evaluates to 0: $2 (\ln 1)^4 = 2 \cdot 0^4 = 0.$

Thus, the definite integral becomes: $2 (\ln 3)^4 - 0 = 2 (\ln 3)^4.$

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Step 4: Compute the Numerical Value

The natural logarithm of 3 is approximately: $\ln 3 \approx 1.0986.$

Raise this to the fourth power: $(\ln 3)^4 = (1.0986)^4 \approx 1.4648.$

Multiply by 2: $2 (\ln 3)^4 = 2 \cdot 1.4648 \approx 2.9296.$

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Final Answer:

$\boxed{2.930 \, \text{(rounded to three decimal places)}}.$

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Do you want further details on this computation, or do you have additional questions?

Related Questions:

1. How does the formula $\int \frac{(\ln x)^n}{x} dx = \frac{(\ln x)^{n+1}}{n+1}$ work?
2. Can this technique be generalized for other values of $n$ in $(\ln x)^n$?
3. What is the significance of the logarithmic term in this integral?
4. How do logarithmic powers affect convergence or divergence in definite integrals?
5. Can this integral be solved using substitution methods?

Tip: Always double-check limits of integration and confirm logarithmic values to avoid