https://file.aiquickdraw.com/mathbot/file/172731445931970z2bqll.jpg

To solve the integral 

\[
\int_3^7 t^4 \ln(4t) \, dt,
\]

we will use **integration by parts**. Integration by parts is based on the formula:

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

Here, we need to choose \( u \) and \( dv \) in a way that simplifies the problem.

Let:

- \( u = \ln(4t) \), so that \( du = \frac{1}{t} \, dt \),
- \( dv = t^4 \, dt \), so that \( v = \frac{t^5}{5} \).

Now, applying the integration by parts formula:

\[
\int_3^7 t^4 \ln(4t) \, dt = \left[ \frac{t^5}{5} \ln(4t) \right]_3^7 - \int_3^7 \frac{t^5}{5} \cdot \frac{1}{t} \, dt.
\]

Simplify the second integral:

\[
\int_3^7 t^4 \, dt = \left[ \frac{t^5}{5} \right]_3^7.
\]

Let’s calculate both terms separately.

### Step 1: Evaluate \( \left[ \frac{t^5}{5} \ln(4t) \right]_3^7 \)
For \( t = 7 \):
\[
\frac{7^5}{5} \ln(4 \cdot 7) = \frac{16807}{5} \ln(28).
\]

For \( t = 3 \):
\[
\frac{3^5}{5} \ln(4 \cdot 3) = \frac{243}{5} \ln(12).
\]

### Step 2: Evaluate \( \int_3^7 t^4 \, dt = \left[ \frac{t^5}{5} \right]_3^7 \)
\[
\frac{7^5}{5} - \frac{3^5}{5} = \frac{16807}{5} - \frac{243}{5} = \frac{16564}{5}.
\]

Now, let’s compute the exact numerical values.

I will perform these calculations.The value of the definite integral

\[
\int_3^7 t^4 \ln(4t) \, dt
\]

is approximately **7767.31**.

Would you like further clarification on any step or more details? 

Here are five related questions for further exploration:
1. How does integration by parts work in general?
2. Can you use integration by parts with different choices of \(u\) and \(dv\)?
3. How does the natural logarithm affect the behavior of the function in this integral?
4. How do definite integrals differ from indefinite integrals in terms of their evaluation?
5. How would you handle an integral like this if the limits were infinite?

**Tip**: When choosing \(u\) in integration by parts, it's often best to pick something that simplifies when differentiated, like a logarithmic or polynomial function.

Evaluate the definite integral ∫ from 3 to 7 of t^4 ln(4t) dt.

Learn how to evaluate the definite integral ∫ from 3 to 7 of t^4 ln(4t) dt using integration by parts. This solution involves detailed steps in calculus, logarithmic functions, and definite integrals, suitable for university-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation