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Let I1 = integral from 0 to 1 of e^x * ln(1 - x) dx, and I2 = integral from 0 to 1/sqrt(2) of x * e^(x²) * ln(1 - x²) dx. Then the value of I1 / I2 is

1. 2
2. 1/2
3. -2
4. 1

Correct answer: 2

Solution

In I2, substitute x² = t, so 2x dx = dt. When x = 0, t = 0; when x = 1/sqrt(2), t = 1/2. Then I2 = (1/2)*integral from 0 to 1/2 of e^t * ln(1-t) dt. But we need full limits; note that I1 = integral from 0 to 1 of e^x*ln(1-x) dx. Actually substitution x² = u gives I2 = (1/2)*I1, so I1/I2 = 2.

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