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人気のある微分積分 >

integral of 6(x+2)ln(x+2)

解

\int 6 (x + 2) ln (x + 2) d x

解

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

解答ステップ

\int 6 (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

= 6 \cdot \int (x + 2) ln (x + 2) d x

拡張

(x + 2) ln (x + 2) : x ln (x + 2) + 2 ln (x + 2)

= 6 \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

= 6 (\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)

= 6 (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

6 (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)) : 6 (\frac{1}{2} x^{2} ln (x + 2) - \frac{1}{4} x^{2} + 2 x ln (x + 2) - x + 2 ln (x + 2) - 1)

= 6 (\frac{1}{2} x^{2} ln (x + 2) - \frac{1}{4} x^{2} + 2 x ln (x + 2) - x + 2 ln (x + 2) - 1)

定数を解答に追加する

= 6 (\frac{1}{2} x^{2} ln (x + 2) - \frac{1}{4} x^{2} + 2 x ln (x + 2) - x + 2 ln (x + 2) - 1) + C

グラフ

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