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人気のある微分積分 >

integral from 0 to 1 of pi(1-x^3)^2

解

\int_{0}^{1} π (1 - x^{3})^{2} d x

解

π \frac{9}{14}

+1

十進法表記

2.01959 \dots

解答ステップ

\int_{0}^{1} π (1 - x^{3})^{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} (1 - x^{3})^{2} d x

拡張

(1 - x^{3})^{2} : 1 - 2 x^{3} + x^{6}

= π \cdot \int_{0}^{1} 1 - 2 x^{3} + x^{6} 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} 1 d x - \int_{0}^{1} 2 x^{3} d x + \int_{0}^{1} x^{6} d x)

\int_{0}^{1} 1 d x = 1

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

\int_{0}^{1} x^{6} d x = \frac{1}{7}

= π (1 - \frac{1}{2} + \frac{1}{7})

結合

1 - \frac{1}{2} + \frac{1}{7} : \frac{9}{14}

= π \frac{9}{14}

グラフ

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