What is the integral from 0 to 1 of x^2e^{-x^2} ?

The integral from 0 to 1 of x^2e^{-x^2} is 0.18947…

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integral from 0 to 1 of x^2e^{-x^2}

\int_{0}^{1} x^{2} e^{- x^{2}} d x

0.18947 \dots

Solution steps

\int_{0}^{1} x^{2} e^{- x^{2}} d x

Simplify

x^{2} : xx

= \int_{0}^{1} 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}^{1}

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

\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}^{1}

Simplify

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

Compute the boundaries:

0.18947 \dots

= 0.18947 \dots

\frac{\partial ^{2}}{\partial x ^{2}} (arctan (\frac{y}{x}))

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

x \to 2 lim (\frac{1}{x ( x ^{2} - 2 )})

\frac{d y}{d x} = \frac{2 x y}{x ^{2} - y ^{2}}, y (1) = 1

\frac{6}{( s ^{3} - 2 ) ^{3}}

What is the integral from 0 to 1 of x^2e^{-x^2} ?
The integral from 0 to 1 of x^2e^{-x^2} is 0.18947…