`int( 1)/(sqrt(9x^(2)-1))dx`

Answer

To solve the integral \( \int \frac{1}{\sqrt{9x^2 - 1}} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We can rewrite the expression under the square root:
\[ 9x^2 - 1 = (3x)^2 - 1^2 \]
Thus, we can express the integral as:
\[ \int \frac{1}{\sqrt{(3x)^2 - 1^2}} \, dx \]

### Step 2: Use the Standard Integral Formula
We will use the standard integral formula:
\[ \int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \ln |x + \sqrt{x^2 - a^2}| + C \]
In our case, we have \( x = 3x \) and \( a = 1 \).

### Step 3: Substitute into the Formula
Applying the formula, we get:
\[ \int \frac{1}{\sqrt{(3x)^2 - 1^2}} \, dx = \ln |3x + \sqrt{(3x)^2 - 1^2}| + C \]

### Step 4: Simplify the Expression
Now, we simplify the expression:
\[ = \ln |3x + \sqrt{9x^2 - 1}| + C \]

### Step 5: Account for the Derivative of the Inner Function
Since we used \( 3x \) instead of \( x \), we need to account for the derivative of \( 3x \) when we integrate:
\[ \int \frac{1}{\sqrt{(3x)^2 - 1^2}} \, dx = \frac{1}{3} \ln |3x + \sqrt{9x^2 - 1}| + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{1}{\sqrt{9x^2 - 1}} \, dx = \frac{1}{3} \ln |3x + \sqrt{9x^2 - 1}| + C \]

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