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integral of (e^{2x})/(\sqrt[3]{1+e^x)}

Lösung

\int \frac{e ^{2 x}}{3 1 + e ^{x}} d x

\frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}} + C

Schritte zur Lösung

\int \frac{e ^{2 x}}{3 1 + e ^{x}} d x

Wende U-Substitution an

= \int \frac{1}{2 3 1 + u} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{1}{3 1 + u} d u

Wende U-Substitution an

= \frac{1}{2} \cdot \int \frac{2 v}{3 1 + v} d v

Entferne die Konstante:

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

= \frac{1}{2} \cdot 2 \cdot \int \frac{v}{3 1 + v} d v

Wende U-Substitution an

= \frac{1}{2} \cdot 2 \cdot \int \frac{w - 1}{3 w} d w

Multipliziere aus

\frac{w - 1}{3 w} : w^{\frac{2}{3}} - \frac{1}{3 w}

= \frac{1}{2} \cdot 2 \cdot \int w^{\frac{2}{3}} - \frac{1}{3 w} d w

Wende die Summenregel an:

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

= \frac{1}{2} \cdot 2 (\int w^{\frac{2}{3}} d w - \int \frac{1}{3 w} d w)

\int w^{\frac{2}{3}} d w = \frac{3}{5} w^{\frac{5}{3}}

\int \frac{1}{3 w} d w = \frac{3}{2} w^{\frac{2}{3}}

= \frac{1}{2} \cdot 2 (\frac{3}{5} w^{\frac{5}{3}} - \frac{3}{2} w^{\frac{2}{3}})

Ersetze zurück

= \frac{1}{2} \cdot 2 (\frac{3}{5} (1 + e^{2 x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{2 x})^{\frac{2}{3}})

Vereinfache

\frac{1}{2} \cdot 2 (\frac{3}{5} (1 + e^{2 x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{2 x})^{\frac{2}{3}}) : \frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}}

= \frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}} + C

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