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人気のある微分積分 >

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

解

\int (x + 2) ln ((x + 2)^{2}) d x

解

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

解答ステップ

\int (x + 2) ln ((x + 2)^{2}) d x

簡素化

ln ((x + 2)^{2}) : 2 ln (x + 2)

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

拡張

ln (x + 2) (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} + 4 x ln (x + 2) - 2 x + 4 ln (x + 2) - 2

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

グラフ

人気の例

integral of 2021

\int 2021

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

\int x (x - 2) (x - 3) (x + 2) (x + 3) d x

limit as x approaches-9 of-8

x \to - 9 lim (- 8)

derivative of x*ln(x^2)

\frac{d}{d x} (x \cdot ln (x^{2}))

(\partial)/(\partial x)((xz)/(y+z))

\frac{\partial}{\partial x} (\frac{x z}{y + z})