Compute in integral `int_(-sqrt(3))^(sqrt(2)) sqrt(4 - x^(2)) dx`

Answer

To compute the integral

\[ \int_{-\sqrt{3}}^{\sqrt{2}} \sqrt{4 - x^2} \, dx, \]

we will use the formula for the integral of the form \(\int \sqrt{a^2 - x^2} \, dx\), which 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 \(a = 2\) in our case.

### Step 1: Identify \(a\) and set up the integral

In our integral, we have \(a^2 = 4\) so \(a = 2\). We can rewrite the integral as:

\[ \int_{-\sqrt{3}}^{\sqrt{2}} \sqrt{4 - x^2} \, dx. \]

### Step 2: Apply the integral formula

Using the formula, we compute:

\[ \int \sqrt{4 - x^2} \, dx = \frac{x}{2} \sqrt{4 - x^2} + \frac{4}{2} \sin^{-1}\left(\frac{x}{2}\right) + C = \frac{x}{2} \sqrt{4 - x^2} + 2 \sin^{-1}\left(\frac{x}{2}\right) + C. \]

### Step 3: Evaluate the definite integral

Now we will evaluate the definite integral from \(-\sqrt{3}\) to \(\sqrt{2}\):

\[ \left[ \frac{x}{2} \sqrt{4 - x^2} + 2 \sin^{-1}\left(\frac{x}{2}\right) \right]_{-\sqrt{3}}^{\sqrt{2}}. \]

### Step 4: Calculate at the upper limit \(x = \sqrt{2}\)

First, we calculate at \(x = \sqrt{2}\):

1. Compute \(\sqrt{4 - (\sqrt{2})^2} = \sqrt{4 - 2} = \sqrt{2}\).
2. Compute \(\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}\).

Now substituting:

\[ \frac{\sqrt{2}}{2} \cdot \sqrt{2} + 2 \cdot \frac{\pi}{4} = 1 + \frac{\pi}{2}. \]

### Step 5: Calculate at the lower limit \(x = -\sqrt{3}\)

Next, we calculate at \(x = -\sqrt{3}\):

1. Compute \(\sqrt{4 - (-\sqrt{3})^2} = \sqrt{4 - 3} = 1\).
2. Compute \(\sin^{-1}\left(\frac{-\sqrt{3}}{2}\right) = -\frac{\pi}{3}\).

Now substituting:

\[ \frac{-\sqrt{3}}{2} \cdot 1 + 2 \cdot \left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} - \frac{2\pi}{3}. \]

### Step 6: Combine the results

Now we combine the results from the upper and lower limits:

\[ \left(1 + \frac{\pi}{2}\right) - \left(-\frac{\sqrt{3}}{2} - \frac{2\pi}{3}\right). \]

This simplifies to:

\[ 1 + \frac{\pi}{2} + \frac{\sqrt{3}}{2} + \frac{2\pi}{3}. \]

### Step 7: Simplify the expression

To combine the \(\pi\) terms, we can find a common denominator:

\[ \frac{3\pi}{6} + \frac{4\pi}{6} = \frac{7\pi}{6}. \]

Thus, the final result is:

\[ 1 + \frac{\sqrt{3}}{2} + \frac{7\pi}{6}. \]

### Final Answer

The value of the integral is:

\[ 1 + \frac{\sqrt{3}}{2} + \frac{7\pi}{6}. \]