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人気のある微分積分 >

integral from 0 to 2 of pi(x^2+2)^2

解

\int_{0}^{2} π (x^{2} + 2)^{2} d x

解

π \frac{376}{15}

+1

十進法表記

78.74925 \dots

解答ステップ

\int_{0}^{2} π (x^{2} + 2)^{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}^{2} (x^{2} + 2)^{2} d x

拡張

(x^{2} + 2)^{2} : x^{4} + 4 x^{2} + 4

= π \cdot \int_{0}^{2} x^{4} + 4 x^{2} + 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

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

\int_{0}^{2} x^{4} d x = \frac{32}{5}

\int_{0}^{2} 4 x^{2} d x = \frac{32}{3}

\int_{0}^{2} 4 d x = 8

= π (\frac{32}{5} + \frac{32}{3} + 8)

結合

\frac{32}{5} + \frac{32}{3} + 8 : \frac{376}{15}

= π \frac{376}{15}

グラフ

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