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人気のある微分積分 >

integral of (x^2-5)^20.2x

解

\int (x^{2} - 5)^{2} 0.2 x d x

解

0.2 (0.16666 \dots x^{6} - 2.5 x^{4} + 12.5 x^{2}) + C

解答ステップ

\int (x^{2} - 5)^{2} \cdot 0.2 x d x

定数を除く:

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

= 0.2 \cdot \int (x^{2} - 5)^{2} x d x

拡張

(x^{2} - 5)^{2} x : x^{5} - 10 x^{3} + 25 x

= 0.2 \cdot \int x^{5} - 10 x^{3} + 25 x 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

= 0.2 (\int x^{5} d x - \int 10 x^{3} d x + \int 25 x d x)

\int x^{5} d x = \frac{x ^{6}}{6}

\int 10 x^{3} d x = \frac{5 x ^{4}}{2}

\int 25 x d x = \frac{25 x ^{2}}{2}

= 0.2 (\frac{x ^{6}}{6} - \frac{5 x ^{4}}{2} + \frac{25 x ^{2}}{2})

簡素化

= 0.2 (0.16666 \dots x^{6} - 2.5 x^{4} + 12.5 x^{2})

定数を解答に追加する

= 0.2 (0.16666 \dots x^{6} - 2.5 x^{4} + 12.5 x^{2}) + C

グラフ

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