calculate-pi-4-pi-4-cosx-e-1-x-1-dx-

Question Number 61662 by maxmathsup by imad last updated on 06/Jun/19

c a l c u l a t e \int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{c o s x}{e^{\frac{1}{x}} + 1} d x

Commented by maxmathsup by imad last updated on 07/Jun/19

l e t f (x) = \frac{c o s x}{e^{\frac{1}{x}} + 1} w e h a v e t h e d e c o m p o s i t i o n f (x) = \frac{f (x) + f (- x)}{2} (e v e n) + \frac{f (x) - f (- x)}{2} (o d d)

\Rightarrow I = \int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{f (x) + f (- x)}{2} d x + \int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{f (x) - f (- x)}{2} d x = H + K

K = 0 \Rightarrow I = \int_{0}^{\frac{π}{4}} {\frac{c o s (x)}{e^{\frac{1}{x}} + 1} + \frac{c o s x}{e^{- \frac{1}{x}} + 1}} d x = \int_{0}^{\frac{π}{4}} {\frac{e^{- \frac{1}{x}} + 1 + e^{\frac{1}{x}} + 1}{1 + e^{\frac{1}{x}} + e^{- \frac{1}{x}} + 1}} c o s x d x

= \int_{0}^{\frac{π}{4}} c o s (x) d x = {[s i n x]}_{0}^{\frac{π}{4}} = \frac{\sqrt{2}}{2} \Rightarrow I = \frac{\sqrt{2}}{2} .