By using the substitution, u = −x^2 evaluate 0∫1 x^3 e^-x^2 dx.

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We have given

I=`int_0^1x^3e^(-x^2)dx`

`u=-x^2`

`du=-2xdx` , x=0 , u=0 , and x=1 ,u=-1

Thus

`I=int_0^{-1}(-u)e^u(-(du)/2)`

`=(1/2)int_0^(-1)ue^udu`

`=(1/2){ue^u-e^u}_0^(-1)`

`=(1/2){(-e^(-1)-e^(-1))-(0-e^0)}`

`=(1/2){-2e^(-1)+1}`

`=(1/2)(1-2e^(-1))`

Here we have applied method of integration by parts .

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