What is the integral of x^ne^{-x^2} ?

The integral of x^ne^{-x^2} is e^{-x^2}x^{n+1}-Undefined+C

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integral of x^ne^{-x^2}

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

e^{- x^{2}} x^{n + 1} - Undefined + C

Solution steps

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

Apply Integration By Parts

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

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

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

Undefined

= e^{- x^{2}} x^{n + 1} - Undefined

Add a constant to the solution

= e^{- x^{2}} x^{n + 1} - Undefined + C

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What is the integral of x^ne^{-x^2} ?
The integral of x^ne^{-x^2} is e^{-x^2}x^{n+1}-Undefined+C