integral from 1 to 3 of (x^3-1)/(x^4)

解

\int_{1}^{3} \frac{x ^{3} - 1}{x ^{4}} d x

ln (3) - \frac{26}{81}

十進法表記

0.77762 \dots

解答ステップ

\int_{1}^{3} \frac{x ^{3} - 1}{x ^{4}} d x

拡張

\frac{x ^{3} - 1}{x ^{4}} : \frac{1}{x} - \frac{1}{x ^{4}}

= \int_{1}^{3} \frac{1}{x} - \frac{1}{x ^{4}} 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}^{3} \frac{1}{x} d x - \int_{1}^{3} \frac{1}{x ^{4}} d x

\int_{1}^{3} \frac{1}{x} d x = ln (3)

\int_{1}^{3} \frac{1}{x ^{4}} d x = \frac{26}{81}

= ln (3) - \frac{26}{81}