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

The integral of (1-2x^2)e^{-x^2} is e^{-x^2}x+C

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integral of (1-2x^2)e^{-x^2}

\int (1 - 2 x^{2}) e^{- x^{2}} d x

e^{- x^{2}} x + C

Solution steps

\int (1 - 2 x^{2}) e^{- x^{2}} d x

Expand

(1 - 2 x^{2}) e^{- x^{2}} : e^{- x^{2}} - 2 e^{- x^{2}} x^{2}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

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

= e^{- x^{2}} x

Add a constant to the solution

= e^{- x^{2}} x + C

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What is the integral of (1-2x^2)e^{-x^2} ?
The integral of (1-2x^2)e^{-x^2} is e^{-x^2}x+C