`int(1)/(sqrt(1-(3x+2)^(2)))dx`

Answer

To solve the integral \( \int \frac{1}{\sqrt{1 - (3x + 2)^2}} \, dx \), we can use the formula for the integral of the form \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1} \left( \frac{x}{a} \right) + C \).

### Step-by-Step Solution:

1. **Identify the Form**:
We need to rewrite the integral in a suitable form. We have:
\[ \int \frac{1}{\sqrt{1 - (3x + 2)^2}} \, dx \]
Here, we can see that \( a^2 = 1 \) and \( x = 3x + 2 \).

2. **Identify \( a \) and \( x \)**:
From the above, we have:
\[ a = 1 \quad \text{and} \quad x = 3x + 2 \]

3. **Differentiate \( x \)**:
We need to find \( \frac{dx}{d(3x + 2)} \):
\[ \frac{dx}{d(3x + 2)} = 3 \quad \Rightarrow \quad dx = 3 \, d(3x + 2) \]

4. **Substitute in the Integral**:
Substitute \( u = 3x + 2 \):
\[ \int \frac{1}{\sqrt{1 - u^2}} \cdot \frac{1}{3} \, du \]

5. **Apply the Integral Formula**:
Using the formula \( \int \frac{1}{\sqrt{1 - u^2}} \, du = \sin^{-1}(u) + C \):
\[ \frac{1}{3} \sin^{-1}(u) + C \]

6. **Back Substitute \( u \)**:
Replace \( u \) back with \( 3x + 2 \):
\[ \frac{1}{3} \sin^{-1}(3x + 2) + C \]

### Final Answer:
Thus, the solution to the integral is:
\[ \int \frac{1}{\sqrt{1 - (3x + 2)^2}} \, dx = \frac{1}{3} \sin^{-1}(3x + 2) + C \]