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人気のある微分積分 >

integral of (11)/(11+e^x)

解

\int \frac{11}{11 + e ^{x}} d x

解

x - ln ∣ e^{x} + 11 ∣ + C

解答ステップ

\int \frac{11}{11 + e ^{x}} d x

定数を除く:

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

= 11 \cdot \int \frac{1}{11 + e ^{x}} d x

u置換積分法を適用する

= 11 \cdot \int \frac{1}{u ( 11 + u )} d u

以下の部分分数を得る:

\frac{1}{u ( 11 + u )} : \frac{1}{11 u} - \frac{1}{11 ( u + 11 )}

= 11 \cdot \int \frac{1}{11 u} - \frac{1}{11 ( u + 11 )} 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

= 11 (\int \frac{1}{11 u} d u - \int \frac{1}{11 ( u + 11 )} d u)

\int \frac{1}{11 u} d u = \frac{1}{11} ln ∣ u ∣

\int \frac{1}{11 ( u + 11 )} d u = \frac{1}{11} ln ∣ u + 11 ∣

= 11 (\frac{1}{11} ln ∣ u ∣ - \frac{1}{11} ln ∣ u + 11 ∣)

代用を戻す

u = e^{x}

= 11 (\frac{1}{11} ln ∣ e^{x} ∣ - \frac{1}{11} ln ∣ e^{x} + 11 ∣)

簡素化

11 (\frac{1}{11} ln ∣ e^{x} ∣ - \frac{1}{11} ln ∣ e^{x} + 11 ∣) : x - ln ∣ e^{x} + 11 ∣

= x - ln ∣ e^{x} + 11 ∣

定数を解答に追加する

= x - ln ∣ e^{x} + 11 ∣ + C

グラフ

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