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Beliebt Rechnen >

integral of 1/(e^x+7)

Lösung

\int \frac{1}{e ^{x} + 7} d x

Lösung

\frac{1}{7} x - \frac{1}{7} ln ∣ e^{x} + 7 ∣ + C

Schritte zur Lösung

\int \frac{1}{e ^{x} + 7} d x

Wende U-Substitution an

= \int \frac{1}{u ( u + 7 )} d u

Ermittle den Partialbruch von

\frac{1}{u ( u + 7 )} : \frac{1}{7 u} - \frac{1}{7 ( u + 7 )}

= \int \frac{1}{7 u} - \frac{1}{7 ( u + 7 )} 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

= \int \frac{1}{7 u} d u - \int \frac{1}{7 ( u + 7 )} d u

\int \frac{1}{7 u} d u = \frac{1}{7} ln ∣ u ∣

\int \frac{1}{7 ( u + 7 )} d u = \frac{1}{7} ln ∣ u + 7 ∣

= \frac{1}{7} ln ∣ u ∣ - \frac{1}{7} ln ∣ u + 7 ∣

Setze in

u = e^{x}

ein

= \frac{1}{7} ln ∣ e^{x} ∣ - \frac{1}{7} ln ∣ e^{x} + 7 ∣

\frac{1}{7} ln ∣ e^{x} ∣ = \frac{1}{7} x

= \frac{1}{7} x - \frac{1}{7} ln ∣ e^{x} + 7 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{7} x - \frac{1}{7} ln ∣ e^{x} + 7 ∣ + C

Graph

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