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人気のある微分積分 >

integral of ((x-1)^2-(x+1)^2)(x^3+1)

解

\int ((x - 1)^{2} - (x + 1)^{2}) (x^{3} + 1) d x

解

- 4 (\frac{x ^{5}}{5} + \frac{x ^{2}}{2}) + C

解答ステップ

\int ((x - 1)^{2} - (x + 1)^{2}) (x^{3} + 1) d x

簡素化

((x - 1)^{2} - (x + 1)^{2}) (x^{3} + 1) : - 4 x (x^{3} + 1)

= \int - 4 x (x^{3} + 1) d x

定数を除く:

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

= - 4 \cdot \int x (x^{3} + 1) d x

拡張

x (x^{3} + 1) : x^{4} + x

= - 4 \cdot \int x^{4} + 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

= - 4 (\int x^{4} d x + \int x d x)

\int x^{4} d x = \frac{x ^{5}}{5}

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

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

定数を解答に追加する

= - 4 (\frac{x ^{5}}{5} + \frac{x ^{2}}{2}) + C

グラフ

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