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Find the area enclosed by the curves y = x e^x, y = x e^(-x), and the vertical line x = 1.

1. 2/e
2. 1 − 2/e
3. 1/e
4. 1 − 1/e

Correct answer: 2/e

Solution

On [0,1], xe^x >= xe^{-x}. Area = integral_0^1 (xe^x - xe^{-x}) dx. With integral xe^x dx = (x-1)e^x and integral xe^{-x} dx = -(x+1)e^{-x}, the first part = 1 and the second = 1 - 2/e. Area = 1 - (1 - 2/e) = 2/e.

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