1-3-e-x-sin-x-x-2-1-dx-pi-12e-

Question Number 138424 by tugu last updated on 13/Apr/21

∣ \underset{1}{\int^{\sqrt{3}}} \frac{e^{- x} s i n x}{x^{2} + 1} d x ∣⩽ \frac{π}{12 e}

Commented by mitica last updated on 14/Apr/21

\exists c \in [1, \sqrt{3}], \underset{1}{\int^{\sqrt{3}}} \frac{e^{- x} s i n x}{x^{2} + 1} d x = \frac{s i n c}{e^{c}} \underset{1}{\int^{\sqrt{3}}} \frac{1}{x^{2} + 1} d x

∣ \frac{s i n c}{e^{c}} \underset{1}{\int^{\sqrt{3}}} \frac{1}{x^{2} + 1} d x ∣⩽ \frac{1}{e^{c}} \cdot ∣ a r c t g x ∣_{1}^{\sqrt{3}} ∣⩽ \frac{1}{e} (\frac{π}{3} - \frac{π}{4}) = \frac{π}{12 e}

Facebook Tweet Pin