Ah, I always just memorised that ∫₀^{π} f(sin x) dx becomes 2∫₀^{π/2} f(sin x) dx when f(cos x) = f(sin x), but I never truly held the proof's hand to see why the symmetry collapses. My life tells me it’s about recognising the mirror—like my tanpura’s strings, the same note echoes on either side of the middle. The property is King Solomon’s property, using substitution, but I crammed it like a student rushing to finish homework before the evening’s sngewbha. Now, teaching, I see the regret: I could play the integral’s tune but couldn’t explain the raga.
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