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Evaluate the integral I = integral from 0 to 10 of ([x]*e^[x])/(e^x - 1) dx, where [x] denotes the greatest integer function (floor of x). The value of I is:

1. 9*(e - 1)
2. 45*(e + 1)
3. 45*(e - 1)
4. 9*(e + 1)

Correct answer: 45*(e - 1)

Solution

Splitting over [n, n+1) and using the substitution, each integral evaluates so that summing n from 1 to 9 gives the factor sum 1+2+...+9=45, with each unit piece contributing (e-1), yielding I = 45*(e-1).

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