`int(3x-2)^(3) dx`

Answer

To solve the integral \( \int (3x - 2)^3 \, dx \), we will follow these steps:

### Step 1: Expand the integrand using the binomial theorem
We start by expanding \( (3x - 2)^3 \) using the binomial expansion formula:
\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]
Here, \( a = 3x \) and \( b = 2 \). Thus, we have:
\[ (3x - 2)^3 = (3x)^3 - 3(3x)^2(2) + 3(3x)(2^2) - (2)^3 \]
Calculating each term:
- \( (3x)^3 = 27x^3 \)
- \( 3(3x)^2(2) = 3 \cdot 9x^2 \cdot 2 = 54x^2 \)
- \( 3(3x)(2^2) = 3 \cdot 3x \cdot 4 = 36x \)
- \( (2)^3 = 8 \)

Putting it all together:
\[ (3x - 2)^3 = 27x^3 - 54x^2 + 36x - 8 \]

### Step 2: Write the integral
Now we can write the integral as:
\[ \int (3x - 2)^3 \, dx = \int (27x^3 - 54x^2 + 36x - 8) \, dx \]

### Step 3: Integrate term by term
Now we will integrate each term separately:
1. \( \int 27x^3 \, dx = 27 \cdot \frac{x^{4}}{4} = \frac{27}{4}x^4 \)
2. \( \int -54x^2 \, dx = -54 \cdot \frac{x^{3}}{3} = -18x^3 \)
3. \( \int 36x \, dx = 36 \cdot \frac{x^{2}}{2} = 18x^2 \)
4. \( \int -8 \, dx = -8x \)

### Step 4: Combine the results
Now we combine all the results of the integrals:
\[ \int (3x - 2)^3 \, dx = \frac{27}{4}x^4 - 18x^3 + 18x^2 - 8x + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final answer is:
\[ \int (3x - 2)^3 \, dx = \frac{27}{4}x^4 - 18x^3 + 18x^2 - 8x + C \]