What is the integral of e^{-x^2}x^2 ?

The integral of e^{-x^2}x^2 is -1/2 e^{-x^2}x+(sqrt(pi))/4 \erf(x)+C

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integral of e^{-x^2}x^2

\int e^{- x^{2}} x^{2} d x

- \frac{1}{2} e^{- x^{2}} x + \frac{π}{4} erf (x) + C

Solution steps

\int e^{- x^{2}} x^{2} d x

Simplify

x^{2} : xx

= \int e^{- x^{2}} xx d x

Apply Integration By Parts

= - \frac{1}{2} e^{- x^{2}} x - \int - \frac{1}{2} e^{- x^{2}} d x

\int - \frac{1}{2} e^{- x^{2}} d x = - \frac{π}{4} erf (x)

= - \frac{1}{2} e^{- x^{2}} x - (- \frac{π}{4} erf (x))

Simplify

= - \frac{1}{2} e^{- x^{2}} x + \frac{π}{4} erf (x)

Add a constant to the solution

= - \frac{1}{2} e^{- x^{2}} x + \frac{π}{4} erf (x) + C

What is the integral of e^{-x^2}x^2 ?
The integral of e^{-x^2}x^2 is -1/2 e^{-x^2}x+(sqrt(pi))/4 \erf(x)+C