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

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

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integral of x^2e^{(-x^2)/2}

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

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

Solution steps

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

Simplify

x^{2} : xx

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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