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

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

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

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

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

Solution steps

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

Simplify

t^{2} : tt

= \int_{0}^{x^{2}} tt e^{t^{2}} d t

Apply Integration By Parts

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

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

\int \frac{1}{2} e^{t^{2}} d t = \frac{π}{4} erfi (t)

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

Compute the boundaries:

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

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

Simplify

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

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