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

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

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

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

- \frac{1}{2} e^{- x^{2}} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int e^{u} d u = e^{u}

= - \frac{1}{2} e^{u}

Substitute back

u = - x^{2}

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

Add a constant to the solution

= - \frac{1}{2} e^{- x^{2}} + C

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