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Evaluate the definite integral: integral from -pi/2 to pi/2 of [x*sin(x) / (1 + e^x)] dx.

1. pi²/4 - 2
2. pi²/4 + 2
3. 1
4. pi² - e^(pi/2)

Correct answer: 1

Solution

By the symmetry property, f(x) + f(-x) = x*sin(x), so the integral equals (1/2)*integral_(-pi/2)^(pi/2) x*sin(x) dx = integral₀^(pi/2) x*sin(x) dx (since x*sin(x) is even). Integrating by parts gives [-x*cos(x) + sin(x)]₀^(pi/2) = 1.

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