By using the properties of definite integrals, evaluate the integrals`int_0^(2) xsqrt(2-x)dx`

Explanation

To evaluate the integral \(\int_0^{2} x \sqrt{2 - x} \, dx\) using properties of definite integrals, we can use a substitution method. Here’s a step-by-step solution:

### Step 1: Set up the substitution
Let \( t = 2 - x \). Then, we differentiate both sides:
\[ dx = -dt \]
Now, we need to change the limits of integration. When \( x = 0 \), \( t = 2 \) and when \( x = 2 \), \( t = 0 \).
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