integral from 0 to 1 of pi(x^4-x^2)

解

\int_{0}^{1} π (x^{4} - x^{2}) d x

- π \frac{2}{15}

十進法表記

- 0.41887 \dots

解答ステップ

\int_{0}^{1} π (x^{4} - x^{2}) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= π \cdot \int_{0}^{1} x^{4} - x^{2} 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

= π (\int_{0}^{1} x^{4} d x - \int_{0}^{1} x^{2} d x)

\int_{0}^{1} x^{4} d x = \frac{1}{5}

\int_{0}^{1} x^{2} d x = \frac{1}{3}

= π (\frac{1}{5} - \frac{1}{3})

簡素化

π (\frac{1}{5} - \frac{1}{3}) : - π \frac{2}{15}

= - π \frac{2}{15}