Stunning Integral!

Calculus Level 5

∫ 0 1 log ⁡ ( Γ ( x ) ) cos ⁡ ( 8 π x ) d x = A B \large \int_0^1 \log (\Gamma(x ))\cos (8\pi x)\, dx= \dfrac {A}{B}

If the equation above holds true for positive integers A A and B B with A , B A,B coprime, find A + B A+B .


The answer is 17.

Aditya Kumar by Aditya Kumar , 22, India

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1 solution

Shivam Sharma
Jun 14, 2017

See my solution... Take n = 4 , we get I = 1/16 , A+B = 17 .

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It would be nice if you explained the first step.

Aditya Kumar - 3 years, 11 months ago

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It's nothing new ... Simply apply the reflection formula of the Gamma function .

( Γ ( x ) ) ( Γ ( 1 − x ) ) = π ( sin ⁡ ( π x ) ) (\Gamma(x))(\Gamma(1-x)) = \frac{\pi}{(\sin(\pi x))} , then put the value of Γ ( x ) \Gamma(x) that's what I had done on the first step .

Shivam Sharma - 3 years, 6 months ago

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