integral from-1 to 0 of (x^{15})/(x^8+1)

解

\int_{- 1}^{0} \frac{x ^{15}}{x ^{8} + 1} d x

- \frac{1}{8} + \frac{1}{8} ln (2)

十進法表記

- 0.03835 \dots

解答ステップ

\int_{- 1}^{0} \frac{x ^{15}}{x ^{8} + 1} d x

長除法

\frac{x ^{15}}{x ^{8} + 1} : x^{7} - \frac{x ^{7}}{x ^{8} + 1}

= \int_{- 1}^{0} x^{7} - \frac{x ^{7}}{x ^{8} + 1} 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}^{0} x^{7} d x - \int_{- 1}^{0} \frac{x ^{7}}{x ^{8} + 1} d x

\int_{- 1}^{0} x^{7} d x = - \frac{1}{8}

\int_{- 1}^{0} \frac{x ^{7}}{x ^{8} + 1} d x = - \frac{1}{8} ln (2)

= - \frac{1}{8} - (- \frac{1}{8} ln (2))

簡素化

= - \frac{1}{8} + \frac{1}{8} ln (2)