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.