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人気のある微分積分 >

integral of 2(x-2)ln(x-2)

解

\int 2 (x - 2) ln (x - 2) d x

解

x^{2} ln (x - 2) - \frac{1}{2} x^{2} + 2 x - 4 x ln (x - 2) + 4 ln (x - 2) - 2 + C

解答ステップ

\int 2 (x - 2) ln (x - 2) d x

定数を除く:

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

= 2 \cdot \int (x - 2) ln (x - 2) d x

拡張

(x - 2) ln (x - 2) : x ln (x - 2) - 2 ln (x - 2)

= 2 \cdot \int x ln (x - 2) - 2 ln (x - 2) 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

= 2 (\int x ln (x - 2) d x - \int 2 ln (x - 2) d x)

\int x ln (x - 2) d x = ln (x - 2) (\frac{1}{2} (x - 2)^{2} + 2 (x - 2)) - \frac{1}{4} (x^{2} + 4 x - 12)

\int 2 ln (x - 2) d x = 2 (x ln (x - 2) - 2 ln (x - 2) - x + 2)

= 2 (ln (x - 2) (\frac{1}{2} (x - 2)^{2} + 2 (x - 2)) - \frac{1}{4} (x^{2} + 4 x - 12) - 2 (x ln (x - 2) - 2 ln (x - 2) - x + 2))

簡素化

2 (ln (x - 2) (\frac{1}{2} (x - 2)^{2} + 2 (x - 2)) - \frac{1}{4} (x^{2} + 4 x - 12) - 2 (x ln (x - 2) - 2 ln (x - 2) - x + 2)) : x^{2} ln (x - 2) - \frac{1}{2} x^{2} + 2 x - 4 x ln (x - 2) + 4 ln (x - 2) - 2

= x^{2} ln (x - 2) - \frac{1}{2} x^{2} + 2 x - 4 x ln (x - 2) + 4 ln (x - 2) - 2

定数を解答に追加する

= x^{2} ln (x - 2) - \frac{1}{2} x^{2} + 2 x - 4 x ln (x - 2) + 4 ln (x - 2) - 2 + C

グラフ

人気の例

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