integral of e^{2x}\sqrt[3]{e^x+2}

Lösung

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

\frac{3}{7} (e^{x} + 2)^{\frac{7}{3}} - \frac{3}{2} (e^{x} + 2)^{\frac{4}{3}} + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int u 3 u + 2 d u

Wende U-Substitution an

= \int (v - 2) 3 v d v

Multipliziere aus

(v - 2) 3 v : v^{\frac{4}{3}} - 2 3 v

= \int v^{\frac{4}{3}} - 2 3 v 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

= \int v^{\frac{4}{3}} d v - \int 2 3 v d v

\int v^{\frac{4}{3}} d v = \frac{3}{7} v^{\frac{7}{3}}

\int 2 3 v d v = \frac{3}{2} v^{\frac{4}{3}}

= \frac{3}{7} v^{\frac{7}{3}} - \frac{3}{2} v^{\frac{4}{3}}

Ersetze zurück

= \frac{3}{7} (e^{x} + 2)^{\frac{7}{3}} - \frac{3}{2} (e^{x} + 2)^{\frac{4}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{7} (e^{x} + 2)^{\frac{7}{3}} - \frac{3}{2} (e^{x} + 2)^{\frac{4}{3}} + C

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