`int(1)/(sqrt(5-.(x^(2))/(4)))dx`

Answer

To solve the integral \( \int \frac{1}{\sqrt{5 - \frac{x^2}{4}}} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the expression under the square root. Notice that:
\[ 5 - \frac{x^2}{4} = 5 - \left(\frac{x}{2}\right)^2 \]
Thus, we can rewrite the integral as:
\[ \int \frac{1}{\sqrt{5 - \left(\frac{x}{2}\right)^2}} \, dx \]

### Step 2: Identify the Form
Now we can identify this integral with the standard form:
\[ \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C \]
Here, we can set \( a = \sqrt{5} \) and \( x = \frac{x}{2} \).

### Step 3: Change of Variables
To apply the formula, we need to change the variable. Let:
\[ u = \frac{x}{2} \quad \Rightarrow \quad x = 2u \quad \Rightarrow \quad dx = 2 \, du \]
Substituting this into the integral gives:
\[ \int \frac{1}{\sqrt{5 - u^2}} \cdot 2 \, du = 2 \int \frac{1}{\sqrt{5 - u^2}} \, du \]

### Step 4: Apply the Standard Integral Formula
Now we can apply the standard integral formula:
\[ 2 \int \frac{1}{\sqrt{5 - u^2}} \, du = 2 \sin^{-1}\left(\frac{u}{\sqrt{5}}\right) + C \]
Substituting back \( u = \frac{x}{2} \):
\[ = 2 \sin^{-1}\left(\frac{\frac{x}{2}}{\sqrt{5}}\right) + C = 2 \sin^{-1}\left(\frac{x}{2\sqrt{5}}\right) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{1}{\sqrt{5 - \frac{x^2}{4}}} \, dx = 2 \sin^{-1}\left(\frac{x}{2\sqrt{5}}\right) + C \]

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