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sweetymoeyz 2022-07-15 Answered
Integrate 0 sin 2 x cosh x + cos x d x x
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Answers (1)

enfeinadag0
Answered 2022-07-16 Author has 16 answers
I'm not sure how you showed the two integrals are equivalent, but the following is an evaluation of
0 e x cos ( x ) tanh ( x ) d x x .
For ( s ) > 0, we have
0 tanh ( t ) e s t d t = 0 1 e 2 t 1 + e 2 t e s t d t = 0 ( 1 e 2 t ) e s t n = 0 ( 1 ) n e 2 t n d t = n = 0 ( 1 ) n 0 ( e ( 2 n + s ) t e ( 2 n + s + 2 ) t ) d t = n = 0 ( 1 ) n 2 n + s n = 0 ( 1 ) n 2 n + s + 2 = 1 2 ( n = 0 ( 1 ) n n + s 2 n = 0 ( 1 ) n n + s 2 + 1 ) = ( 1 ) 1 4 ( ψ ( s 4 + 1 2 ) ψ ( s 4 ) ψ ( s 4 + 1 ) + ψ ( s 4 + 1 2 ) ) = ( 2 ) 1 2 ( ψ ( s 4 + 1 2 ) ψ ( s 4 ) 2 s ) .
Therefore,
0 e x cos ( x ) tanh ( x ) d x x = 0 e ( 1 + i ) x tanh ( x ) x d x = 0 e ( 1 + i ) x tanh ( x ) 0 e x t d t d x = 0 0 tanh ( x ) e ( 1 + i + t ) x d x d t = 0 1 2 ( ψ ( 1 + i + t 4 + 1 2 ) ψ ( 1 + i + t 4 ) 2 1 + i + t ) d t = ( 2 ln Γ ( 3 + i + t 4 ) 2 ln Γ ( 1 + i + t 4 ) ln ( 1 + i + t ) ) | 0 = ( 3 ) ( 2 ln ( 2 ) 2 ln Γ ( 3 + i 4 ) + 2 ln Γ ( 1 + i 4 ) + ln ( 1 + i ) ) = 3 ln ( 2 ) 2 + 2 ( ln Γ ( 1 + i 4 ) ln Γ ( 3 + i 4 ) ) .

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