`intsqrt(4-x^(2))dx`

Answer

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

### Step 1: Identify the form of the integral
We recognize that the integral \(\sqrt{4 - x^2}\) can be related to the standard integral form \(\int \sqrt{a^2 - x^2} \, dx\), where \(a = 2\).

### Step 2: Use the standard formula
The standard formula for the integral \(\int \sqrt{a^2 - x^2} \, dx\) is given by:
\[ \int \sqrt{a^2 - x^2} \, dx = \frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) + C \]
where \(C\) is the constant of integration.

### Step 3: Substitute \(a = 2\)
In our case, \(a = 2\). We can substitute this value into the formula:
\[ \int \sqrt{4 - x^2} \, dx = \frac{x}{2} \sqrt{4 - x^2} + \frac{4}{2} \sin^{-1}\left(\frac{x}{2}\right) + C \]

### Step 4: Simplify the expression
Now, we simplify the expression:
\[ \int \sqrt{4 - x^2} \, dx = \frac{x}{2} \sqrt{4 - x^2} + 2 \sin^{-1}\left(\frac{x}{2}\right) + C \]

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

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