`int_(0)^(sqrt(2))sqrt(2-x^(2))dx=?`

Options

A. `pi`
B. `2pi`
C. `(pi)/(2)`
D. none of these

Explanation

To solve the integral \( \int_{0}^{\sqrt{2}} \sqrt{2 - x^2} \, dx \), we can use a trigonometric substitution. Here’s a step-by-step solution:

### Step 1: Trigonometric Substitution
We can use the substitution \( x = \sqrt{2} \sin \theta \). Then, the differential \( dx \) becomes:
\[ dx = \sqrt{2} \cos \theta \, d\theta \]
Next, we need to change the limits of integration. When \( x = 0 \), \( \theta = 0 \), and when \( x = \sqrt{2} \), \( \theta = \frac{\pi}{2} \).
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