https://file.aiquickdraw.com/mathbot/file/17271042744544eouccln.jpg

The integral in the image is:

\[
\int_{2}^{3} \frac{12x - 1}{\sqrt[3]{3x - 1}} \, dx
\]

Let's solve this step by step.

### Step 1: Substitution
To simplify this integral, we can use substitution. Let:

\[
u = 3x - 1
\]

Thus, \( du = 3 \, dx \) or \( dx = \frac{du}{3} \).

When \( x = 2 \), \( u = 3(2) - 1 = 5 \), and when \( x = 3 \), \( u = 3(3) - 1 = 8 \).

Now, rewrite the integral in terms of \( u \):

\[
\int_{5}^{8} \frac{12x - 1}{\sqrt[3]{u}} \cdot \frac{du}{3}
\]

### Step 2: Express \( x \) in terms of \( u \)
Since \( u = 3x - 1 \), we can solve for \( x \):

\[
x = \frac{u + 1}{3}
\]

Now substitute this into \( 12x - 1 \):

\[
12x - 1 = 12 \left( \frac{u + 1}{3} \right) - 1 = 4(u + 1) - 1 = 4u + 4 - 1 = 4u + 3
\]

### Step 3: Rewrite the integral
Substitute this into the integral:

\[
\int_{5}^{8} \frac{4u + 3}{\sqrt[3]{u}} \cdot \frac{du}{3}
\]

This simplifies to:

\[
\frac{1}{3} \int_{5}^{8} \frac{4u + 3}{\sqrt[3]{u}} \, du
\]

### Step 4: Split the integral
We can split the integral into two parts:

\[
\frac{1}{3} \left( \int_{5}^{8} \frac{4u}{\sqrt[3]{u}} \, du + \int_{5}^{8} \frac{3}{\sqrt[3]{u}} \, du \right)
\]

These integrals can now be simplified.

1. \(\frac{4u}{\sqrt[3]{u}} = 4u^{\frac{2}{3}}\)
2. \(\frac{3}{\sqrt[3]{u}} = 3u^{-\frac{1}{3}}\)

### Step 5: Solve each integral
Now integrate each part:

1. \(\int u^{\frac{2}{3}} \, du = \frac{3}{5} u^{\frac{5}{3}}\)
2. \(\int u^{-\frac{1}{3}} \, du = \frac{3}{2} u^{\frac{2}{3}}\)

Now substitute back into the original expression:

\[
\frac{1}{3} \left( 4 \cdot \frac{3}{5} u^{\frac{5}{3}} + 3 \cdot \frac{3}{2} u^{\frac{2}{3}} \right)
\]

Simplify this:

\[
\frac{1}{3} \left( \frac{12}{5} u^{\frac{5}{3}} + \frac{9}{2} u^{\frac{2}{3}} \right)
\]

### Step 6: Evaluate the definite integrals
Now we evaluate this from \( u = 5 \) to \( u = 8 \).

1. For \( \frac{12}{5} u^{\frac{5}{3}} \):

\[
\frac{12}{5} \left( 8^{\frac{5}{3}} - 5^{\frac{5}{3}} \right)
\]

2. For \( \frac{9}{2} u^{\frac{2}{3}} \):

\[
\frac{9}{2} \left( 8^{\frac{2}{3}} - 5^{\frac{2}{3}} \right)
\]

Let me calculate these values.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, the steps I outlined for solving the integral are valid, and you can now evaluate the expressions manually or with a calculator. Would you like help with any specific part of the explanation?

Here are some related questions you could explore:
1. How do you choose substitutions in integrals involving roots?
2. What is the geometric interpretation of integrals?
3. How does substitution help simplify complex integrals?
4. Why are fractional powers used in these types of integrals?
5. What are other common techniques for solving definite integrals?

**Tip**: When evaluating complex integrals, breaking them down into simpler parts like I did can make the calculation easier!

Calculate the integral from 2 to 3 of (12x - 1) divided by the cube root of (3x - 1) dx.

This problem involves calculating a definite integral from 2 to 3 of (12x - 1) / cube root(3x - 1). Learn how to solve this using substitution and the properties of fractional powers. Suitable for calculus students.

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