We are asked to compute the integral:

$\int_{3}^{6} 4x^3 \ln(x^4) \, dx$

Step-by-step solution:

1. Simplify the logarithmic expression: Recall that $\ln(a^b) = b \ln(a)$. In this case, $\ln(x^4) = 4\ln(x)$.

   Thus, the integral becomes: $\int_{3}^{6} 4x^3 \cdot 4\ln(x) \, dx = 16 \int_{3}^{6} x^3 \ln(x) \, dx$

2. Use integration by parts: To solve $\int x^3 \ln(x) \, dx$, we'll use the integration by parts formula:

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

   Here, let:

   • $u = \ln(x)$, so $du = \frac{1}{x} dx$
   • $dv = x^3 dx$, so $v = \frac{x^4}{4}$

3. Apply the integration by parts formula: Substituting into the formula: $\int x^3 \ln(x) \, dx = \frac{x^4}{4} \ln(x) - \int \frac{x^4}{4} \cdot \frac{1}{x} \, dx$ Simplify the second term: $= \frac{x^4}{4} \ln(x) - \frac{1}{4} \int x^3 \, dx$

4. Integrate $x^3$: $\int x^3 \, dx = \frac{x^4}{4}$

5. Substitute back into the equation: $\int x^3 \ln(x) \, dx = \frac{x^4}{4} \ln(x) - \frac{x^4}{16}$

6. Multiply by 16: Now, multiply the result by 16, since the original problem included this factor: $16 \int x^3 \ln(x) \, dx = 4x^4 \ln(x) - x^4$

7. Evaluate the definite integral from 3 to 6: Now, evaluate the expression at the limits 3 and 6: $\left[ 4x^4 \ln(x) - x^4 \right]_{3}^{6}$

   At $x = 6$: $4(6^4) \ln(6) - 6^4 = 4(1296) \ln(6) - 1296 = 5184 \ln(6) - 1296$

   At $x = 3$: $4(3^4) \ln(3) - 3^4 = 4(81) \ln(3) - 81 = 324 \ln(3) - 81$

8. Final result: Subtract the two evaluations: $(5184 \ln(6) - 1296) - (324 \ln(3) - 81)$ $= 5184 \ln(6) - 324 \ln(3) - 1296 + 81$ $= 5184 \ln(6) - 324 \ln(3) - 1215$

Thus, the value of the integral is:

$5184 \ln(6) - 324 \ln(3) - 1215$

If you'd like, I can also compute the numerical approximation of this expression.

Would you like more details on any step or a numerical answer?

Here are 5 related questions to expand on the topic:

1. How do you determine when to apply integration by parts?
2. What are common logarithmic properties that simplify integrals?
3. Can integration by parts be applied iteratively? When does it help?
4. How would this integral change if the limits of integration were altered?
5. How do definite integrals represent areas under curves?

Tip: Always check if the integral can be simplified algebraically before applying more advanced techniques like integration by parts.