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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Simplify

x^{2} : xx

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

Apply Integration By Parts

= - 2 (- \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)

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

Simplify

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

= e^{- x^{2}} x - \frac{π}{2} erf (x)

Add a constant to the solution

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

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