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人気のある微分積分 >

integral from 2 to 3 of pi(x+x^2)^2

解

\int_{2}^{3} π (x + x^{2})^{2} d x

解

π \frac{2431}{30}

+1

十進法表記

254.57372 \dots

解答ステップ

\int_{2}^{3} π (x + x^{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_{2}^{3} (x + x^{2})^{2} d x

拡張

(x + x^{2})^{2} : x^{2} + 2 x^{3} + x^{4}

= π \cdot \int_{2}^{3} x^{2} + 2 x^{3} + 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

= π (\int_{2}^{3} x^{2} d x + \int_{2}^{3} 2 x^{3} d x + \int_{2}^{3} x^{4} d x)

\int_{2}^{3} x^{2} d x = \frac{19}{3}

\int_{2}^{3} 2 x^{3} d x = \frac{65}{2}

\int_{2}^{3} x^{4} d x = \frac{211}{5}

= π (\frac{19}{3} + \frac{65}{2} + \frac{211}{5})

結合

\frac{19}{3} + \frac{65}{2} + \frac{211}{5} : \frac{2431}{30}

= π \frac{2431}{30}

グラフ

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