高数基础题
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∫(0->+∞) arctanx / (1+x^2)^(3/2)dx
let
x= tana
dx= (seca)^2 da
x=0,a=0
x=+∞,a=π/2
∫(0->+∞) arctanx / (1+x^2)^(3/2)dx
=∫(0->π/2) [a / (seca)^3 ] (seca)^2 da
=∫(0->π/2) acosa da
=∫(0->π/2) adsina
=[asina](0->π/2) - ∫(0->π/2)sina da
= π/2+[cosa](0->π/2)
= π/2-1
let
x= tana
dx= (seca)^2 da
x=0,a=0
x=+∞,a=π/2
∫(0->+∞) arctanx / (1+x^2)^(3/2)dx
=∫(0->π/2) [a / (seca)^3 ] (seca)^2 da
=∫(0->π/2) acosa da
=∫(0->π/2) adsina
=[asina](0->π/2) - ∫(0->π/2)sina da
= π/2+[cosa](0->π/2)
= π/2-1
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