`I= int (dx)/( sqrt( 5-x^(2) - 4x) )`.

Answer

To solve the integral \( I = \int \frac{dx}{\sqrt{5 - x^2 - 4x}} \), we will follow these steps:

### Step 1: Simplify the expression under the square root
We start with the expression under the square root:
\[ 5 - x^2 - 4x \]
We can rearrange this to make it easier to complete the square. We rewrite it as:
\[ 5 - (x^2 + 4x) \]
Now, we complete the square for \( x^2 + 4x \):
\[ x^2 + 4x = (x + 2)^2 - 4 \]
Thus, we can rewrite the expression under the square root:
\[ 5 - (x^2 + 4x) = 5 - ((x + 2)^2 - 4) = 5 + 4 - (x + 2)^2 = 9 - (x + 2)^2 \]

### Step 2: Rewrite the integral
Now we can substitute this back into the integral:
\[ I = \int \frac{dx}{\sqrt{9 - (x + 2)^2}} \]

### Step 3: Use the standard integral formula
We recognize that this integral is of the form:
\[ \int \frac{dx}{\sqrt{a^2 - u^2}} = \sin^{-1}\left(\frac{u}{a}\right) + C \]
where \( a = 3 \) and \( u = x + 2 \). Thus, we can apply the formula:
\[ I = \sin^{-1}\left(\frac{x + 2}{3}\right) + C \]

### Final Answer
Therefore, the solution to the integral is:
\[ I = \sin^{-1}\left(\frac{x + 2}{3}\right) + C \]