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The value of the definite integral I = integral from -pi/4 to pi/4 of [ 1 / ((1 + e^(x*cos(x))) * (sin⁴(x) + cos⁴(x))) ] dx is equal to:

1. -pi/2
2. pi / (2 * sqrt(2))
3. -pi/4
4. pi / sqrt(2)

Correct answer: pi / (2 * sqrt(2))

Solution

After eliminating the exponential by symmetry, I reduces to the integral of 1/(sin⁴x+cos⁴x) from 0 to pi/4. Substituting t=tan(x) and using (1+t²)/(1+t⁴) with the u=t-1/t trick yields pi/(2*sqrt(2)).

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