integral from 0 to 2 of x(e^{x^2}-e^4)

解

\int_{0}^{2} x (e^{x^{2}} - e^{4}) d x

\frac{e ^{4} - 1}{2} - e^{4} \cdot 2

十進法表記

- 82.39722 \dots

解答ステップ

\int_{0}^{2} x (e^{x^{2}} - e^{4}) d x

拡張

x (e^{x^{2}} - e^{4}) : e^{x^{2}} x - e^{4} x

= \int_{0}^{2} e^{x^{2}} x - e^{4} x d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{2} e^{x^{2}} x d x - \int_{0}^{2} e^{4} x d x

\int_{0}^{2} e^{x^{2}} x d x = \frac{e ^{4} - 1}{2}

\int_{0}^{2} e^{4} x d x = e^{4} \cdot 2

= \frac{e ^{4} - 1}{2} - e^{4} \cdot 2