0-ln-cosh-x-cosh-x-dx-0-pi-2-log-1-sin-x-dx-

Question Number 141419 by mnjuly1970 last updated on 18/May/21

ϕ := \int_{0}^{\infty} \frac{l n (c o s h (x))}{c o s h (x)} d x \overset{?}{=} \int_{0}^{\frac{π}{2}} l o g (\frac{1}{s i n (x)}) d x

Answered by mindispower last updated on 18/May/21

t = c h (t)

= \int_{1}^{\infty} \frac{l n (t)}{t} . \frac{d t}{\sqrt{t^{2} - 1}}

= \int_{1}^{\infty} \frac{l n (t)}{t^{2} \sqrt{1 - \frac{1}{t^{2}}}} d t

l e t \frac{1}{t} = s i n (y)

= \int_{\frac{π}{2}}^{0} \frac{l n (\frac{1}{s i n (y)})}{\sqrt{1 - {s i n}^{2} (y)}} . - c o s (y) d y

= \int_{0}^{\frac{π}{2}} l n (\frac{1}{s i n (y)}) d y

Commented by mindispower last updated on 18/May/21

p l e a s u r

Commented by mnjuly1970 last updated on 18/May/21

t h a n k s a l o t s i r p o w e r . .

Answered by mnjuly1970 last updated on 18/May/21

:= \frac{1}{2} \int_{0}^{\infty} \frac{c o s h (x) l n (1 + {s i n h}^{2} (x))}{{c o s h}^{2} (x)} d x

: \overset{s i n h (x) = t}{=} \frac{1}{2} \int_{0}^{\infty} \frac{l n (1 + t^{2})}{1 + t^{2}} d t

: \overset{t = t a n (y)}{=} \frac{1}{2} \int_{0}^{\frac{π}{2}} \frac{l n (1 + {t a n}^{2} (y))}{1} d y

:= \frac{1}{2} \int_{0}^{\frac{π}{2}} l n {(\frac{1}{c o s (y)})}^{2} d y

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

\dots \dots \dots \dots ϕ := \int_{0}^{\frac{π}{2}} l n (\frac{1}{s i n (x)}) d x = \frac{π}{2} l n (2) \dots \dots \dots .

\dots \dots \dots . m . n . j u l y . 1970 \dots \dots \dots .