Interesting Integral
The equation above holds true for coprime integers and . Find .
The answer is 2014.
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1 solution
I = ∫ 0 2 π cos 2 0 1 1 x sin 2 0 1 3 x d x = ∫ 0 2 π cos 2 0 1 1 x ( sin x cos 2 0 1 2 x + cos x sin 2 0 1 2 x ) d x = ∫ 0 2 π cos 2 0 1 1 x sin x cos 2 0 1 2 x d x + ∫ 0 2 π cos 2 0 1 2 x sin 2 0 1 2 x d x = − 2 0 1 2 cos 2 0 1 2 x cos 2 0 1 2 x ∣ ∣ ∣ ∣ 0 2 π − ∫ 0 2 π cos 2 0 1 2 x sin 2 0 1 2 x d x + ∫ 0 2 π cos 2 0 1 2 x sin 2 0 1 2 x d x = 2 0 1 2 1 By integration by parts
⟹ 2 p + q = 2 + 2 0 1 2 = 2 0 1 4
Generalization: ∫ 0 2 π cos n x sin ( n + 2 ) x d x = n + 1 1