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人気のある微分積分 >

integral from 0 to 1 of (8x^2-2)e^x

解

\int_{0}^{1} (8 x^{2} - 2) e^{x} d x

解

2 (3 e - 7)

+1

十進法表記

2.30969 \dots

解答ステップ

\int_{0}^{1} (8 x^{2} - 2) e^{x} d x

(8 x^{2} - 2) e^{x} = 2 (4 x^{2} - 1) e^{x}

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

定数を除く:

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

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

部分積分を適用する

= 2 [(4 x^{2} - 1) e^{x} - \int 8 x e^{x} d x]_{0}^{1}

\int 8 x e^{x} d x = 8 (e^{x} x - e^{x})

= 2 [(4 x^{2} - 1) e^{x} - 8 (e^{x} x - e^{x})]_{0}^{1}

拡張

(4 x^{2} - 1) e^{x} - 8 (e^{x} x - e^{x}) : 4 e^{x} x^{2} - 8 e^{x} x + 7 e^{x}

= 2 [4 e^{x} x^{2} - 8 e^{x} x + 7 e^{x}]_{0}^{1}

限度を計算する:

3 e - 7

= 2 (3 e - 7)

グラフ

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