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

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

Solution steps

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

Simplify

x^{3} : x^{2} x

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

\int \frac{25}{x} d x

\frac{d}{d x} (\frac{x ^{2} - 5 x}{x})

\int \frac{1}{121 - x ^{2}} d x

\frac{d}{d x} (\frac{x ^{2} + 1}{x + 4})

7 x^{2} y^{^{''}} + 15 x y^{^{'}} + y = 0

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