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

The integral from 0 to x^2 of x^2e^{x^2} is (e^{x^4}x^2)/2-(sqrt(pi))/4 \erfi(x^2)

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

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

\frac{e ^{x^{4}} x ^{2}}{2} - \frac{π}{4} erfi (x^{2})

Solution steps

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

Simplify

x^{2} : xx

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

Simplify

x^{2} : xx

= \int_{0}^{xx} 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}^{x^{2}}

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

Compute the boundaries:

\frac{1}{2} e^{x^{4}} x^{2} - \frac{π}{4} erfi (x^{2})

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

Simplify

= \frac{e ^{x^{4}} x ^{2}}{2} - \frac{π}{4} erfi (x^{2})

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What is the integral from 0 to x^2 of x^2e^{x^2} ?
The integral from 0 to x^2 of x^2e^{x^2} is (e^{x^4}x^2)/2-(sqrt(pi))/4 \erfi(x^2)