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

1. e*(sum from n=1 to 9 of n)
2. 9e
3. 45*(e-1)
4. e¹⁰ - 1

Correct answer: 45*(e-1)

Solution

Breaking [0,10] into subintervals and integrating each gives I = sumₙ₌₀⁹ n * integralₙⁿ⁺¹ e^(n+1-x) dx = sumₙ₌₀⁹ n*(e-1) = (e-1)*sumₙ₌₀⁹ n = 45*(e-1).

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