What is the integral of 2/9 xe^{-(x/3)^2} ?

The integral of 2/9 xe^{-(x/3)^2} is -e^{-(x^2)/9}+C

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integral of 2/9 xe^{-(x/3)^2}

\int \frac{2}{9} x e^{- (\frac{x}{3})^{2}} d x

- e^{- \frac{x ^{2}}{9}} + C

Solution steps

\int \frac{2}{9} x e^{- (\frac{x}{3})^{2}} d x

Take the constant out:

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

= \frac{2}{9} \cdot \int x e^{- (\frac{x}{3})^{2}} d x

Simplify

x e^{- (\frac{x}{3})^{2}} : e^{- \frac{x ^{2}}{9}} x

= \frac{2}{9} \cdot \int e^{- \frac{x ^{2}}{9}} x d x

Apply u-substitution

: \int - \frac{9 e ^{u}}{2} d u

= \frac{2}{9} \cdot \int - \frac{9 e ^{u}}{2} d u

Take the constant out:

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

= \frac{2}{9} (- \frac{9}{2} \cdot \int e^{u} d u)

Use the common integral:

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

= \frac{2}{9} (- \frac{9}{2} e^{u})

Substitute back

u = - \frac{x ^{2}}{9}

= \frac{2}{9} (- \frac{9}{2} e^{- \frac{x ^{2}}{9}})

Simplify

\frac{2}{9} (- \frac{9}{2} e^{- \frac{x ^{2}}{9}}) : - e^{- \frac{x ^{2}}{9}}

= - e^{- \frac{x ^{2}}{9}}

Add a constant to the solution

= - e^{- \frac{x ^{2}}{9}} + C

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What is the integral of 2/9 xe^{-(x/3)^2} ?
The integral of 2/9 xe^{-(x/3)^2} is -e^{-(x^2)/9}+C