设f(x)=s(1,x)cos(t^2)dt.求设s(0,1)f(x)dx.望赐教!s表定积分符号,s(0,1)即为从0至1的定积分.

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设f(x)=s(1,x)cos(t^2)dt.求设s(0,1)f(x)dx.望赐教!s表定积分符号,s(0,1)即为从0至1的定积分.
设f(x)=s(1,x)cos(t^2)dt.求设s(0,1)f(x)dx.望赐教!
s表定积分符号,s(0,1)即为从0至1的定积分.

设f(x)=s(1,x)cos(t^2)dt.求设s(0,1)f(x)dx.望赐教!s表定积分符号,s(0,1)即为从0至1的定积分.
f '(x)=cos(x^2)
∫[0-->1] f(x)dx=[0-->1] xf(x)-∫[0-->1] xf '(x)dx
=f(1)-∫[0-->1] xcos(x^2)dx 注意f(1)=0
=-1/2∫[0-->1] cos(x^2)d(x^2)
=-1/2sin(x^2) [0-->1]
=-1/2sin1

将积分次序换一下即可
s(0,1)f(x)dx=s(0,1)[s(1,x) cos(t^2)dt]dx
=s(1,0)[s(0,t)cos(t^2)dx]dt
=s(1,0)[tcos(t^2)]dt
=1/2 sin(t^2) |(1,0)
=-(sin1)/2

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