integral from 1/2 to 1 of x(e-e^x)

解

\int_{\frac{1}{2}}^{1} x (e - e^{x}) d x

\frac{3 e}{8} - \frac{e}{2}

解答ステップ

\int_{\frac{1}{2}}^{1} x (e - e^{x}) d x

拡張

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

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

\int_{\frac{1}{2}}^{1} e x d x = \frac{3 e}{8}

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

= \frac{3 e}{8} - \frac{e}{2}