prove-that-0-e-cos-x-sin-sin-x-x-dx-pi-2-e-1-

Question Number 141062 by mnjuly1970 last updated on 15/May/21

p r o v e t h a t ::

ϕ := \int_{0}^{\infty} \frac{e^{c o s (x)} s i n (s i n (x))}{x} d x = \frac{π}{2} (e - 1)

Commented by mindispower last updated on 15/May/21

a l w a y s p l e a s u r h a v e n i c e d a y “ A \overset{. .}{i d} {M o b a r a k}^{″}

Commented by mnjuly1970 last updated on 15/May/21

t h a n k y o u s o m u c h s i r

p o w e r y o u r A i d m o b a r a k . .

Commented by mindispower last updated on 15/May/21

s t a r t e w i t h e y o u r p r e v i o u s r e s u l t q u a t i o n

\sum_{n ⩾ 1} a^{n} \frac{s i n (n x)}{n!} = e^{a c o s (x)} s i n (a s i n (x)

{ϕ = \int_{0}^{\infty} \frac{1}{x} \sum_{n ⩾ 1} (a^{n} s i n (n x)) / (n!))}_{a = 1} d x

= \sum_{n ⩾ 1} \int_{0}^{\infty} \frac{s i n (n x)}{x (n!)} d x = \sum_{n ⩾ 1} \frac{1}{n!} \int_{0}^{\infty} \frac{s i n (n x)}{n x} d (n x)

= \sum_{n ⩾ 1} \frac{1}{n!} . \frac{π}{2} = \frac{π}{2} (\sum_{n ⩾ 0} \frac{1}{n!} - 1) = \frac{π}{2} (e - 1)

Commented by mnjuly1970 last updated on 15/May/21

g r a t e f u l m r p o w e r \dots .

Answered by Dwaipayan Shikari last updated on 15/May/21

e^{c o s (x)} s i n (s i n x) = Σ \frac{s i n (n x)}{n!}

\int_{0}^{\infty} \sum_{n = 1} \frac{s i n (n x)}{x n!} = \frac{π}{2} \underset{n = 1}{\sum^{\infty}} \frac{1}{n!} = \frac{π}{2} (e - 1)

Commented by mnjuly1970 last updated on 15/May/21

m e r c e y m r P a y a n \dots