pi-2-pi-2-dx-1-e-sin-x-

Question Number 89946 by jagoll last updated on 20/Apr/20

\int \underset{- \frac{π}{2}}{\overset{\frac{π}{2}}{}} \frac{dx}{1 + e^{\sin x}}

Answered by john santu last updated on 20/Apr/20

I = \underset{\frac{- π}{2}}{\int^{\frac{π}{2}}} (\frac{d x}{1 + e^{\sin x}})

r e p l a c e x b y - x

I = \underset{\frac{π}{2}}{\int^{- \frac{π}{2}}} (\frac{- d x}{1 + e^{- \sin x}}) = \underset{- \frac{π}{2}}{\int^{\frac{π}{2}}} (\frac{d x}{1 + e^{- \sin x}})

I + I = \underset{- \frac{π}{2}}{\int^{\frac{π}{2}}} \frac{1 + e^{\sin x}}{1 + e^{\sin x}} d x

2 I = \underset{- \frac{π}{2}}{\int^{\frac{π}{2}}} 1 d x = π

I = \frac{π}{2}