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人気のある微分積分 >

integral of (2v)/(7v-12)

解

\int \frac{2 v}{7 v - 12} d v

解

\frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣) + C

解答ステップ

\int \frac{2 v}{7 v - 12} d v

定数を除く:

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

= 2 \cdot \int \frac{v}{7 v - 12} d v

u置換積分法を適用する

= 2 \cdot \int \frac{u + 12}{49 u} d u

定数を除く:

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

= 2 \cdot \frac{1}{49} \cdot \int \frac{u + 12}{u} d u

拡張

\frac{u + 12}{u} : 1 + \frac{12}{u}

= 2 \cdot \frac{1}{49} \cdot \int 1 + \frac{12}{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

= 2 \cdot \frac{1}{49} (\int 1 d u + \int \frac{12}{u} d u)

\int 1 d u = u

\int \frac{12}{u} d u = 12 ln ∣ u ∣

= 2 \cdot \frac{1}{49} (u + 12 ln ∣ u ∣)

代用を戻す

u = 7 v - 12

= 2 \cdot \frac{1}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣)

簡素化

2 \cdot \frac{1}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣) : \frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣)

= \frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣)

定数を解答に追加する

= \frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣) + C

グラフ

人気の例

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\int (x^{2} + 1)^{10} d x

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