Explanation
1. Understanding the Exponent
The term in the exponent is x−[x], which is the fractional part of x, denoted as {x}. This function behaves very differently from a standard linear x.
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On the interval [0,1), {x}=x.
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On the interval [1,2), {x}=x−1.
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This pattern repeats every integer.
2. The Property of Periodic Functions
The function e{x} is periodic with period T=1. A fundamental property of definite integrals for periodic functions is:
Applying this to your problem where n=1000 and T=1:
∫01000e{x}dx=1000∫01e{x}dx
3. Calculating the Single Unit
Between 0 and 1, the fractional part {x} is simply x.
∫01exdx=[ex]01=e1−e0=e−1
4. Conclusion
Multiplying the result of one period by the total number of periods (1000):