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




Question Number 138424 by tugu last updated on 13/Apr/21
∣∫_1 ^(√3)  ((e^(−x) sin x)/(x^2 +1))dx∣≤(π/(12e))
31exsinxx2+1dx∣⩽π12e
Commented by mitica last updated on 14/Apr/21
∃c∈[1,(√3)],∫_1 ^(√3) ((e^(−x) sinx)/(x^2 +1))dx=((sinc)/e^c ) ∫_1 ^(√3) (1/(x^2 +1))dx  ∣((sinc)/e^c ) ∫_1 ^(√3) (1/(x^2 +1))dx∣≤(1/e^c )∙∣arctgx∣_1 ^(√3) ∣≤(1/e)((π/3)−(π/4))=(π/(12e))
c[1,3],31exsinxx2+1dx=sincec311x2+1dxsincec311x2+1dx∣⩽1ecarctgx13∣⩽1e(π3π4)=π12e

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