de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of 8/(8+e^x)

Lösung

\int \frac{8}{8 + e ^{x}} d x

Lösung

x - ln ∣ e^{x} + 8 ∣ + C

Schritte zur Lösung

\int \frac{8}{8 + e ^{x}} d x

Entferne die Konstante:

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

= 8 \cdot \int \frac{1}{8 + e ^{x}} d x

Wende U-Substitution an

= 8 \cdot \int \frac{1}{u ( 8 + u )} d u

Ermittle den Partialbruch von

\frac{1}{u ( 8 + u )} : \frac{1}{8 u} - \frac{1}{8 ( u + 8 )}

= 8 \cdot \int \frac{1}{8 u} - \frac{1}{8 ( u + 8 )} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 8 (\int \frac{1}{8 u} d u - \int \frac{1}{8 ( u + 8 )} d u)

\int \frac{1}{8 u} d u = \frac{1}{8} ln ∣ u ∣

\int \frac{1}{8 ( u + 8 )} d u = \frac{1}{8} ln ∣ u + 8 ∣

= 8 (\frac{1}{8} ln ∣ u ∣ - \frac{1}{8} ln ∣ u + 8 ∣)

Setze in

u = e^{x}

ein

= 8 (\frac{1}{8} ln ∣ e^{x} ∣ - \frac{1}{8} ln ∣ e^{x} + 8 ∣)

Vereinfache

8 (\frac{1}{8} ln ∣ e^{x} ∣ - \frac{1}{8} ln ∣ e^{x} + 8 ∣) : x - ln ∣ e^{x} + 8 ∣

= x - ln ∣ e^{x} + 8 ∣

Füge eine Konstante zur Lösung hinzu

= x - ln ∣ e^{x} + 8 ∣ + C

Graph

Beliebte Beispiele

steigung y^2=9x,\at x=16

s l o p e y^{2} = 9 x, at x = 16

limit as x approaches-3 of sqrt(4x+28-4)

x \to - 3 lim (4 x + 28 - 4)

derivative of (7sqrt(x)/(2x^3))

\frac{d}{d x} (\frac{7 x}{2 x ^{3}})

f(x)=1-\sqrt[3]{x^2}

f (x) = 1 - 3 x^{2}

integral of (2x+5)^2

\int (2 x + 5)^{2} d x