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Beliebt Rechnen >

integral of (34)/(34+e^x)

Lösung

\int \frac{34}{34 + e ^{x}} d x

Lösung

x - ln ∣ e^{x} + 34 ∣ + C

Schritte zur Lösung

\int \frac{34}{34 + e ^{x}} d x

Entferne die Konstante:

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

= 34 \cdot \int \frac{1}{34 + e ^{x}} d x

Wende U-Substitution an

= 34 \cdot \int \frac{1}{u ( 34 + u )} d u

Ermittle den Partialbruch von

\frac{1}{u ( 34 + u )} : \frac{1}{34 u} - \frac{1}{34 ( u + 34 )}

= 34 \cdot \int \frac{1}{34 u} - \frac{1}{34 ( u + 34 )} 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

= 34 (\int \frac{1}{34 u} d u - \int \frac{1}{34 ( u + 34 )} d u)

\int \frac{1}{34 u} d u = \frac{1}{34} ln ∣ u ∣

\int \frac{1}{34 ( u + 34 )} d u = \frac{1}{34} ln ∣ u + 34 ∣

= 34 (\frac{1}{34} ln ∣ u ∣ - \frac{1}{34} ln ∣ u + 34 ∣)

Setze in

u = e^{x}

ein

= 34 (\frac{1}{34} ln ∣ e^{x} ∣ - \frac{1}{34} ln ∣ e^{x} + 34 ∣)

Vereinfache

34 (\frac{1}{34} ln ∣ e^{x} ∣ - \frac{1}{34} ln ∣ e^{x} + 34 ∣) : x - ln ∣ e^{x} + 34 ∣

= x - ln ∣ e^{x} + 34 ∣

Füge eine Konstante zur Lösung hinzu

= x - ln ∣ e^{x} + 34 ∣ + C

Graph

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