积分(0,pie) xsinx/(1+(cosx)^2) dx
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积分(0,pie) xsinx/(1+(cosx)^2) dx
积分范围为从零到pie.
积分范围为从零到pie.
let f(x) = xsinx/(1+(cosx)^2
f(-x) = f(x)
ie f(x) is even function
∫(0,π)xsinx/(1+(cosx)^2) dx = ∫(-π,0)xsinx/(1+(cosx)^2) dx
I= ∫(0,π)xsinx/(1+(cosx)^2) dx (1)
let y =(π-x)
dy = dx
x=0,y=π
x=π ,y=0
I= ∫(π,0)(π-y)(siny) /(1+(cosy)^2) (-dy)
= -∫(0,π)(π-y)siny/(1+(cosy)^2)dy
= -∫(0,π)(π-x)sinx/(1+(cosx)^2)dx
I= -∫(0,π)πsinx/(1+(cosx)^2)dx + I
=> ∫(0,π)πsinx/(1+(cosx)^2)dx =0
let y' =-(π-x)
dy' = dx
x=0,y'=-π
x=π ,y' =0
I= ∫(-π,0)-(π-y')(-siny') /(1+(cosy')^2) (dy')
= ∫(-π,0)(π-y')siny/(1+(cosy')^2)dy'
= ∫(-π,0)(π-x)sinx/(1+(cosx)^2)dx
= ∫(-π,0)πsinx/(1+(cosx)^2)dx -I
2I = ∫(-π,0)πsinx/(1+(cosx)^2)dx
let g(x) = πsinx/(1+(cosx)^2)dx
g(-x) = -g(x) ( g is odd)
∫(-π,0)πsinx/(1+(cosx)^2)dx = -∫(0,π)πsinx/(1+(cosx)^2)dx =0
=> I= ∫(0,π)xsinx/(1+(cosx)^2) dx =0
f(-x) = f(x)
ie f(x) is even function
∫(0,π)xsinx/(1+(cosx)^2) dx = ∫(-π,0)xsinx/(1+(cosx)^2) dx
I= ∫(0,π)xsinx/(1+(cosx)^2) dx (1)
let y =(π-x)
dy = dx
x=0,y=π
x=π ,y=0
I= ∫(π,0)(π-y)(siny) /(1+(cosy)^2) (-dy)
= -∫(0,π)(π-y)siny/(1+(cosy)^2)dy
= -∫(0,π)(π-x)sinx/(1+(cosx)^2)dx
I= -∫(0,π)πsinx/(1+(cosx)^2)dx + I
=> ∫(0,π)πsinx/(1+(cosx)^2)dx =0
let y' =-(π-x)
dy' = dx
x=0,y'=-π
x=π ,y' =0
I= ∫(-π,0)-(π-y')(-siny') /(1+(cosy')^2) (dy')
= ∫(-π,0)(π-y')siny/(1+(cosy')^2)dy'
= ∫(-π,0)(π-x)sinx/(1+(cosx)^2)dx
= ∫(-π,0)πsinx/(1+(cosx)^2)dx -I
2I = ∫(-π,0)πsinx/(1+(cosx)^2)dx
let g(x) = πsinx/(1+(cosx)^2)dx
g(-x) = -g(x) ( g is odd)
∫(-π,0)πsinx/(1+(cosx)^2)dx = -∫(0,π)πsinx/(1+(cosx)^2)dx =0
=> I= ∫(0,π)xsinx/(1+(cosx)^2) dx =0
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