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

Answer

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

### Step 1: Simplify the expression under the square root
First, we simplify the expression \( x^2 + 4x - 4 \):
\[ x^2 + 4x - 4 = (x + 2)^2 - 8 \]
Thus, we can rewrite the integral as:
\[ \int \frac{dx}{x \sqrt{(x + 2)^2 - 8}} \]

### Step 2: Use substitution
Next, we will use the substitution:
\[ x + 2 = 2\sqrt{2} \sin(\theta) \quad \Rightarrow \quad dx = 2\sqrt{2} \cos(\theta) d\theta \]
This gives us:
\[ x = 2\sqrt{2} \sin(\theta) - 2 \]

### Step 3: Substitute in the integral
Substituting \( x \) and \( dx \) into the integral, we have:
\[ \sqrt{(x + 2)^2 - 8} = \sqrt{(2\sqrt{2} \sin(\theta))^2 - 8} = \sqrt{8\sin^2(\theta) - 8} = 2\sqrt{2} \sqrt{\sin^2(\theta) - 1} = 2\sqrt{2} \cos(\theta) \]
Thus, the integral becomes:
\[ \int \frac{2\sqrt{2} \cos(\theta) d\theta}{(2\sqrt{2} \sin(\theta) - 2)(2\sqrt{2} \cos(\theta))} \]

### Step 4: Simplify the integral
Now, simplifying the integral:
\[ \int \frac{2\sqrt{2} \cos(\theta) d\theta}{(2\sqrt{2} \sin(\theta) - 2)(2\sqrt{2} \cos(\theta))} = \int \frac{d\theta}{2\sqrt{2} \sin(\theta) - 2} \]
This can be simplified further:
\[ = \frac{1}{2\sqrt{2}} \int \frac{d\theta}{\sin(\theta) - 1} \]

### Step 5: Solve the integral
The integral \( \int \frac{d\theta}{\sin(\theta) - 1} \) can be solved using partial fractions or other methods, leading to:
\[ \int \frac{d\theta}{\sin(\theta) - 1} = -\ln|\tan(\theta/2)| + C \]

### Step 6: Back-substitution
Now, we need to back-substitute \( \theta \) in terms of \( x \):
From our substitution \( x + 2 = 2\sqrt{2} \sin(\theta) \), we have:
\[ \sin(\theta) = \frac{x + 2}{2\sqrt{2}} \]
Thus, we substitute back to get:
\[ \int \frac{dx}{x \sqrt{x^2 + 4x - 4}} = -\frac{1}{2\sqrt{2}} \ln\left| \tan\left(\frac{1}{2} \sin^{-1}\left(\frac{x + 2}{2\sqrt{2}}\right) \right) \right| + C \]

### Final Answer
The final answer for the integral is:
\[ -\frac{1}{2} \sin^{-1}\left(\frac{x + 2}{2\sqrt{2}}\right) + C \]