To evaluate the integral 

\[
\int_{3}^{7} t^3 \ln(3t) \, dt,
\]

we can use the method of integration by parts. Let's set it up as follows:

1. Choose \( u = \ln(3t) \) and \( dv = t^3 \, dt \).

   - Then \( du = \frac{1}{3t} \cdot 3 \, dt = \frac{1}{t} \, dt \).
   - And \( v = \int t^3 \, dt = \frac{t^4}{4} \).

Using the integration by parts formula:

\[
\int u \, dv = uv - \int v \, du,
\]

we substitute:

\[
u = \ln(3t), \quad dv = t^3 \, dt, \quad du = \frac{1}{t} \, dt, \quad v = \frac{t^4}{4}.
\]

Thus, the integral becomes:

\[
\int_{3}^{7} t^3 \ln(3t) \, dt = \left[ \frac{t^4}{4} \ln(3t) \right]_{3}^{7} - \int_{3}^{7} \frac{t^4}{4} \cdot \frac{1}{t} \, dt.
\]

Simplifying the integral:

\[
= \left[ \frac{t^4}{4} \ln(3t) \right]_{3}^{7} - \frac{1}{4} \int_{3}^{7} t^3 \, dt.
\]

Now, we evaluate each term separately:

1. **Evaluate \( \left[ \frac{t^4}{4} \ln(3t) \right]_{3}^{7} \):**

   \[
   \left[ \frac{7^4}{4} \ln(21) \right] - \left[ \frac{3^4}{4} \ln(9) \right].
   \]

   Calculate:

   \[
   = \frac{2401}{4} \ln(21) - \frac{81}{4} \ln(9).
   \]

2. **Evaluate \( \frac{1}{4} \int_{3}^{7} t^3 \, dt \):**

   \[
   = \frac{1}{4} \left[ \frac{t^4}{4} \right]_{3}^{7} = \frac{1}{16} \left[ 7^4 - 3^4 \right].
   \]

   Calculate:

   \[
   = \frac{1}{16} \left[ 2401 - 81 \right] = \frac{1}{16} \times 2320 = 145.
   \]

Finally, combining these results:

\[
= \frac{2401}{4} \ln(21) - \frac{81}{4} \ln(9) - 145.
\]

This is the evaluated form of the integral:

\[
\int_{3}^{7} t^3 \ln(3t) \, dt = \frac{2401}{4} \ln(21) - \frac{81}{4} \ln(9) - 145.
\]

Would you like more details, or do you have any questions? Here are some related questions to consider:

1. How do you choose \( u \) and \( dv \) in integration by parts?
2. What are other methods for integrating functions involving logarithms?
3. How would the process change if the integral was from 0 to infinity?
4. Can integration by parts be used more than once on a single integral?
5. What is the physical significance of logarithmic functions in real-world applications?

**Tip:** Integration by parts is often useful when the integrand is a product of a polynomial and a logarithmic or exponential function.

Learn how to evaluate the integral ∫[3 to 7] t^3 ln(3t) dt using integration by parts with a step-by-step solution. This problem involves advanced calculus concepts including integration techniques and logarithmic functions. Suitable for advanced college-level students and calculus enthusiasts.

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