帮忙求个定积分,
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帮忙求个定积分,
帮忙求个定积分,
帮忙求个定积分,
∫(0,1)1/3y^3e^(-y^2)dy
=1/3∫(0,1)y^2*y*e^(-y^2)dy
=1/3∫(0,1)y^2e^(-y^2)dy^2/2
=-1/6∫(0,1)y^2e^(-y^2)d(-y^2)
=-1/6∫(0,1)y^2de^(-y^2)
=-1/6(e^(-y^2)*y^2-∫(0,1)e^(-y^2)dy^2)
=-1/6(e^(-y^2)*y^2+∫(0,1)de^(-y^2))
=-1/6(e^(-y^2)*y^2+e^(-y^2)+C ) (0,1)
=-1/6*e^(-y^2)(y^2+1) +C) (0,1)
=-1/6(e^-1*(1+1)+C-e^0*(0+1)-C)
=-1/6(2/e-1)
∫1/3 *y^3*e^(-y^2)dy
=∫1/6 *y^2*e^(-y^2)d(y^2)
=∫1/6 *x*e^(-x)dx (令x=y^2,积分区域不变,还是从0到1)
= - ∫1/6 *x*de^(-x)
= -1/6* [xe^(-x)- ∫e^(-x)dx]
=-1/6* [xe^(-x)- e^(-x)+C]
计算上式在1和0处的差值即可
(0,1)∫(1/3)y³e^(-y²)dy
先作不定积分 ∫(1/3)y³e^(-y²)dy=-(1/6)∫y²d[e^(-y²)]=-(1/6)[y²e^(-y²)-∫e^(-y²)d(y²)]
=-(1/6)[y²e^(-y²)+∫e^(-y²)d...
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(0,1)∫(1/3)y³e^(-y²)dy
先作不定积分 ∫(1/3)y³e^(-y²)dy=-(1/6)∫y²d[e^(-y²)]=-(1/6)[y²e^(-y²)-∫e^(-y²)d(y²)]
=-(1/6)[y²e^(-y²)+∫e^(-y²)d(-y²)]=-(1/6)[y²e^(-y²)+e^(-y²)]=-(1/6)e^(-y²)(y²+1)(积分常数省去未写)。
故定积分(0,1)∫(1/3)y³e^(-y²)dy=-(1/6){[e^(-y²)](y²+1)}(0,1)=-(1/6)[(2/e)-1]
收起
令x=y^2
原式=1/6∫[0,1] xe^(-x)dx
=-1/6∫[0,1] xd[e^(-x)]
然后用分部积分