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Evaluate I = integral from 0 to 10 of [x] * e^([x] - x + 1) dx, where [x] denotes the greatest integer function (floor function). Express the answer in simplified form.

1. (e-1) * (sum from n=1 to 9 of n) = 45*(e-1)
2. 9*(e-1)
3. 45*(e-1)
4. e - 1

Correct answer: 45*(e-1)

Solution

I = sum from n=0 to 9 of integral from n to n+1 of n*e^(n-x+1) dx. For n=0: integrand=0, contributes 0. For n>=1: integralₙ^(n+1) n*e^(n+1-x)dx = n*[-e^(n+1-x)]ₙ^(n+1) = n*(-e⁰ + e¹) = n*(e-1). Summing: I = (e-1)*sum(n=0 to 9 of n) = (e-1)*(0+1+2+...+9) = (e-1)*45 = 45*(e-1).

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