Integration problem: int ln (sin sqrt(x)+ cos sqrt(x))dx

Jack Ingram 2022-10-24 Answered
Integration problem: ln ( sin ( x ) + cos ( x ) ) d x
I really don't know how to start solving this problem; any tips or solutions will be greatly appreciated.
Thank you in advance.
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Answers (1)

Cagliusov8
Answered 2022-10-25 Author has 15 answers
By setting x = u 2 we have:
I = 2 u log ( sin u + cos u ) d u = u 2 log ( sin u + cos u ) u 2 1 tan u 1 + tan u d u
and, by putting v = π 4 v
u 2 1 tan u 1 + tan u d u = ( π 4 v ) 2 tan v d v
Now we may exploit tan v d v = log cos v, so the last integral just depends on:
v log ( cos v ) d v
that, however, is not an elementary function, but a combination of a logarithm, a dilogarithm and a trilogarithm multiplied by powers of v: just write cos v as e i v + e i v 2 , exploit the Taylor series of log ( 1 + z ) and integrate termwise.
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