What is the integral of x^2e^{-1/2 x^2} ?

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

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integral of x^2e^{-1/2 x^2}

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

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

Solution steps

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

Simplify

- \frac{1}{2} x^{2} : - \frac{x ^{2}}{2}

= \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} erf (\frac{x}{2})

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

Simplify

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

Add a constant to the solution

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

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