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

Answer

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

### Step 1: Expand the expression \( (3x - 2)^3 \)
We can use the binomial expansion formula for \( (a - b)^3 \):
\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]
Here, let \( a = 3x \) and \( b = 2 \). Thus,
\[ (3x - 2)^3 = (3x)^3 - 3(3x)^2(2) + 3(3x)(2^2) - 2^3 \]
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: Set up the integral
Now we substitute the expanded expression back into the integral:
\[ \int (3x - 2)^3 \, dx = \int (27x^3 - 54x^2 + 36x - 8) \, dx \]

### Step 3: Integrate each term
We will integrate term by term:
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. \( \int -8 \, dx = -8x \)

Putting these results together:
\[ \int (27x^3 - 54x^2 + 36x - 8) \, dx = \frac{27}{4}x^4 - 18x^3 + 18x^2 - 8x + C \]

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