Step 1: Evaluate the integral I = ∫_0^∞ x³e^(−x)dx. Step 2: Compare with the Gamma definition Γ(z) = ∫_0^∞ t^(z−1)e^(−t)dt. Here the exponent of x is 3, so z − 1 = 3, giving z = 4. Step 3: Therefore I = Γ(4). Step 4: Use the recurrence Γ(n) = (n−1)! for integers: Γ(4) = 3! = 6. Step 5: Check via repeated integration by parts: ∫x³e^(−x)dx reduces through x², x, and constant terms, each step bringing down the exponent — confirming 6. Result: I = 6. The same method gives ∫_0^∞ x^(1/2)e^(−x)dx = Γ(3/2) = √π/2.