What is the integral from 0 to 2 of x^2e^{-x^2} ?

The integral from 0 to 2 of x^2e^{-x^2} is 0.42272…

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integral from 0 to 2 of x^2e^{-x^2}

\int_{0}^{2} x^{2} e^{- x^{2}} d x

0.42272 \dots

Solution steps

\int_{0}^{2} x^{2} e^{- x^{2}} d x

Simplify

x^{2} : xx

= \int_{0}^{2} xx e^{- x^{2}} d x

Apply Integration By Parts

: [- \frac{1}{2} e^{- x^{2}} x - \int - \frac{1}{2} e^{- x^{2}} d x]_{0}^{2}

= [- \frac{1}{2} e^{- x^{2}} x - \int - \frac{1}{2} e^{- x^{2}} d x]_{0}^{2}

\int - \frac{1}{2} e^{- x^{2}} d x = - \frac{π}{4} erf (x)

= [- \frac{1}{2} e^{- x^{2}} x - (- \frac{π}{4} erf (x))]_{0}^{2}

Simplify

= [- \frac{1}{2} e^{- x^{2}} x + \frac{π}{4} erf (x)]_{0}^{2}

Compute the boundaries:

0.42272 \dots

= 0.42272 \dots

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What is the integral from 0 to 2 of x^2e^{-x^2} ?
The integral from 0 to 2 of x^2e^{-x^2} is 0.42272…