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人気のある微分積分 >

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

解

\int \frac{3 x}{3 x + 2} d x

解

x + \frac{2}{3} - \frac{2}{3} ln ∣ 3 x + 2 ∣ + C

解答ステップ

\int \frac{3 x}{3 x + 2} d x

定数を除く:

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

= 3 \cdot \int \frac{x}{3 x + 2} d x

u置換積分法を適用する

= 3 \cdot \int \frac{u - 2}{9 u} d u

定数を除く:

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

= 3 \cdot \frac{1}{9} \cdot \int \frac{u - 2}{u} d u

拡張

\frac{u - 2}{u} : 1 - \frac{2}{u}

= 3 \cdot \frac{1}{9} \cdot \int 1 - \frac{2}{u} d u

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

= 3 \cdot \frac{1}{9} (\int 1 d u - \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= 3 \cdot \frac{1}{9} (u - 2 ln ∣ u ∣)

代用を戻す

u = 3 x + 2

= 3 \cdot \frac{1}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣)

簡素化

3 \cdot \frac{1}{9} (3 x + 2 - 2 ln ∣ 3 x + 2 ∣) : x + \frac{2}{3} - \frac{2}{3} ln ∣ 3 x + 2 ∣

= x + \frac{2}{3} - \frac{2}{3} ln ∣ 3 x + 2 ∣

定数を解答に追加する

= x + \frac{2}{3} - \frac{2}{3} ln ∣ 3 x + 2 ∣ + C

グラフ

人気の例

limit as x approaches 3 of (x^2-25)/(sqrt(x^2+11))

x \to 3 lim (\frac{x ^{2} - 25}{x ^{2} + 11})

integral from 0 to 2 of 2e^{-3x}

\int_{0}^{2} 2 e^{- 3 x} d x

derivative of (x^{1/3}-1^3)

\frac{d}{d x} ((x^{\frac{1}{3}} - 1)^{3})

xy^'=y+3x^2sin(x),y(pi)=0

x y^{^{'}} = y + 3 x^{2} sin (x), y (π) = 0

integral from 2 to 4 of 1/(x^2-25)

\int_{2}^{4} \frac{1}{x ^{2} - 25} d x