`int(dx)/(sqrt(9x^(2)-4))`

Answer

To solve the integral \( \int \frac{dx}{\sqrt{9x^2 - 4}} \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{dx}{\sqrt{9x^2 - 4}} \]

### Step 2: Factor Out the Constant
We notice that we can factor out a constant from the square root:
\[ 9x^2 - 4 = 9\left(x^2 - \frac{4}{9}\right) \]
Thus, we can rewrite the integral as:
\[ I = \int \frac{dx}{\sqrt{9} \sqrt{x^2 - \frac{4}{9}}} = \frac{1}{3} \int \frac{dx}{\sqrt{x^2 - \left(\frac{2}{3}\right)^2}} \]

### Step 3: Use the Standard Integral Formula
We recognize that the integral \( \int \frac{dx}{\sqrt{x^2 - a^2}} \) has a known result:
\[ \int \frac{dx}{\sqrt{x^2 - a^2}} = \ln\left| x + \sqrt{x^2 - a^2} \right| + C \]
In our case, \( a = \frac{2}{3} \).

### Step 4: Apply the Formula
Now we can apply the formula to our integral:
\[ I = \frac{1}{3} \left( \ln\left| x + \sqrt{x^2 - \left(\frac{2}{3}\right)^2} \right| + C \right) \]
Substituting \( a \) back in:
\[ I = \frac{1}{3} \ln\left| x + \sqrt{x^2 - \frac{4}{9}} \right| + C \]

### Step 5: Final Answer
Thus, the final answer for the integral is:
\[ \int \frac{dx}{\sqrt{9x^2 - 4}} = \frac{1}{3} \ln\left| x + \sqrt{x^2 - \frac{4}{9}} \right| + C \]

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