Integral-I-0-pi-ln-sin-x-tan-1-cot-x-dx-0-proof-I-0-pi-ln-sin-x-tan-1-tan-pi-2-x-dx-0-pi-pi-2-x-

Question Number 152201 by mnjuly1970 last updated on 26/Aug/21

\dots Integral \dots

I := \int_{0}^{π} l n (s i n (x)) . {t a n}^{- 1} (c o t (x)) d x \overset{?}{=} 0

p r o o f :: \dots .

I := \int_{0}^{π} l n (s i n (x)) . {t a n}^{- 1} (t a n (\frac{π}{2} - x)) d x

:= \int_{0}^{π} (\frac{π}{2} - x) . l n (s i n (x)) d x

:= \frac{π}{2} \int_{0}^{π} l n (s i n (x)) d x - \int_{0}^{π} x l n (s i n (x)) d x

:= \frac{π}{2} (- π l n (2)) - J \dots \dots (1)

J : = \int_{0}^{π} (π - x) l n (s i n (x)) d x

:= π (- π l n (2)) - J

∴ J := \frac{- π^{2}}{2} l n (2) \dots \dots . (2)

(2) ⇛ (1) : I = 0 \dots \dots \dots