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

解

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

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

解答ステップ

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

簡素化

t^{2} : tt

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

部分積分を適用する

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

\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}

限度を計算する:

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

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

簡素化

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