Integrate int(log(sin x cos x))^n dx with hypergeometric function form

clealtAfforcewug

clealtAfforcewug

Answered question

2022-11-18

Integrate ( log ( sin x cos x ) ) n d x with hypergeometric function form
Could anyone help me with that?

Answer & Explanation

Ismael Wilkinson

Ismael Wilkinson

Beginner2022-11-19Added 13 answers

Hint:
( log ( sin x cos x ) ) n   d x = ( log sin 2 x 2 ) n   d x
Let u = log sin 2 x 2
Then x = sin 1 ( 2 e u ) 2
d x = e u 1 4 e 2 u d u
( log sin 2 x 2 ) n   d x
= u n e u 1 4 e 2 u d u
Case 1: | 4 e 2 u | 1
Then u n e u 1 4 e 2 u d u
= m = 0 ( 2 m ) ! u n e ( 2 m + 1 ) u ( m ! ) 2 d u
= u n 2 i 1 e 2 u 4 d u
= m = 0 i ( 2 m ) ! u n e 2 m u 2 4 m + 1 ( m ! ) 2 d u
= m = 0 i ( 2 m ) ! u n e 2 m u 2 4 m + 1 ( m ! ) 2 d u
= m = 0 k = 0 i ( 1 ) k + 1 ( 2 m ) ! m k u n + k 2 4 m k + 1 ( m ! ) 2 k ! d u

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