What is `int x*e^(-x^2) dx`

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`int xe^(-x^2) dx= -1/2 int -2xe^(-x^2) dx= -1/2 int d e^(-x^2)=-1/2 e^(-x^2)`

We have to find the value of : `int x*e^(-x^2) dx`

Let -x^2 = y

dy/dx = -2x

x*dx = (-1/2)*dy

`int x*e^(-x^2) dx`

=> `int (-1/2)*e^y dy`

=> `(-1/2)*e^y`

substitute y = -x^2

=> `-1/(2*e^(x^2))`

**The required integral is** `-1/(2*e^(x^2))`

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