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Evaluate the definite integral ∫₀¹⁰⁰⁰ e^(x-x) dx.

1. e¹⁰⁰⁰-1
2. (e¹⁰⁰⁰-1)/(e-1)
3. 1000(e-1)
4. (e-1)/(1000)

Correct answer: 1000(e-1)

Solution

The exponent is the fractional part x-[x], periodic with period 1. Over each unit interval integral of e^(x-[x]) equals integral_0^1 e^t dt = e-1. Over [0,1000] there are 1000 such intervals, so the value is 1000(e-1).

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