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

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

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

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

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

Solution steps

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

Simplify

= \int e^{- \frac{x ^{2}}{2}} x^{3} d x

Apply u-substitution

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

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

Apply Integration By Parts

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

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

\int - 2 e^{- \frac{u}{2}} d u = 4 e^{- \frac{u}{2}}

= \frac{1}{2} (- 2 e^{- \frac{u}{2}} u - 4 e^{- \frac{u}{2}})

Substitute back

u = x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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