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Beliebt Rechnen >

integral of sqrt(e^x-1)

Lösung

\int e^{x} - 1 d x

Lösung

2 (- arctan (e^{x} - 1) + e^{x} - 1) + C

Schritte zur Lösung

\int e^{x} - 1 d x

Wende U-Substitution an

= \int \frac{u - 1}{u} d u

Wende U-Substitution an

= \int \frac{2 v ^{2}}{v ^{2} + 1} d v

Entferne die Konstante:

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

= 2 \cdot \int \frac{v ^{2}}{v ^{2} + 1} d v

\frac{v ^{2}}{v ^{2} + 1} = - \frac{1}{v ^{2} + 1} + 1

= 2 \cdot \int - \frac{1}{v ^{2} + 1} + 1 d v

Wende die Summenregel an:

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

= 2 (- \int \frac{1}{v ^{2} + 1} d v + \int 1 d v)

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

\int 1 d v = v

= 2 (- arctan (v) + v)

Ersetze zurück

= 2 (- arctan (e^{x} - 1) + e^{x} - 1)

Füge eine Konstante zur Lösung hinzu

= 2 (- arctan (e^{x} - 1) + e^{x} - 1) + C

Graph

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