integral from 1 to e of (x^5+1)/x

解

\int_{1}^{e} \frac{x ^{5} + 1}{x} d x

\frac{e ^{5} - 1}{5} + 1

十進法表記

30.48263 \dots

解答ステップ

\int_{1}^{e} \frac{x ^{5} + 1}{x} d x

拡張

\frac{x ^{5} + 1}{x} : x^{4} + \frac{1}{x}

= \int_{1}^{e} x^{4} + \frac{1}{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_{1}^{e} x^{4} d x + \int_{1}^{e} \frac{1}{x} d x

\int_{1}^{e} x^{4} d x = \frac{e ^{5} - 1}{5}

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

= \frac{e ^{5} - 1}{5} + 1