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Question:

If f(x) = (2 - x cos x) / (2 + x cos x) and g(x) = log_e x (x > 0), then the value of integral ∫_{-π/4}^{π/4} g(f(x)) dx is:

log_e 2

log_e 3

log_e e

log_e 1

Solution:

Correct option is D. log_e 1.

g(f(x)) = ln(f(x)) = ln((2 - x cos x) / (2 + x cos x)).
Therefore, I = ∫_{-π/4}^{π/4} ln((2 - x cos x) / (2 + x cos x)) dx = ∫_0^{π/4} (ln((2 - x cos x) / (2 + x cos x)) + ln((2 + x cos x) / (2 - x cos x))) dx = ∫_0^{π/4} (0) dx = 0 = log_e (1).