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 e^4-(sqrt(pi))/4 \erfi(2)

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

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

e^{4} - \frac{π}{4} erfi (2)

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} erfi (x)

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

Compute the boundaries:

e^{4} - \frac{π}{4} erfi (2)

= e^{4} - \frac{π}{4} erfi (2)

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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 e^4-(sqrt(pi))/4 \erfi(2)