How to solve this equation for y ? and is it an irrational number or a rational? 2

Brunton39

Brunton39

Answered question

2022-06-15

How to solve this equation for y? and is it an irrational number or a rational?
2 7 π 2 log 2 + 16 7 0 π 2 x log ( sin x ) d x y π 3 = 0.

Answer & Explanation

Cahokiavv

Cahokiavv

Beginner2022-06-16Added 31 answers

Assuming that the logarithm has base b, the integral will be equal to:
16 7 0 π 2 x log ( sin x ) d x = π 2 ln ( 4 ) 7 ζ ( 3 ) 7 ln ( b ) ,where ζ ( 3 ) - is the Riemann zeta function. So
2 7 π 2 log 2 + 16 7 0 π 2 x log ( sin x ) d x = ζ ( 3 ) ln ( b )
And
y = ( ζ ( 3 ) ln ( b ) π 3 ) 2 Buy the way ζ ( 3 ) is proved to be irrational.

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