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Beliebt Rechnen >

integral of (e^{2x})/(\sqrt[4]{e^x+1)}

Lösung

\int \frac{e ^{2 x}}{4 e ^{x} + 1} d x

Lösung

\frac{4}{7} (e^{x} + 1)^{\frac{7}{4}} - \frac{4}{3} (e^{x} + 1)^{\frac{3}{4}} + C

Schritte zur Lösung

\int \frac{e ^{2 x}}{4 e ^{x} + 1} d x

Wende U-Substitution an

= \int \frac{u - 1}{4 u} d u

Multipliziere aus

\frac{u - 1}{4 u} : u^{\frac{3}{4}} - \frac{1}{4 u}

= \int u^{\frac{3}{4}} - \frac{1}{4 u} 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 u^{\frac{3}{4}} d u - \int \frac{1}{4 u} d u

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

\int \frac{1}{4 u} d u = \frac{4}{3} u^{\frac{3}{4}}

= \frac{4}{7} u^{\frac{7}{4}} - \frac{4}{3} u^{\frac{3}{4}}

Setze in

u = e^{x} + 1

ein

= \frac{4}{7} (e^{x} + 1)^{\frac{7}{4}} - \frac{4}{3} (e^{x} + 1)^{\frac{3}{4}}

Füge eine Konstante zur Lösung hinzu

= \frac{4}{7} (e^{x} + 1)^{\frac{7}{4}} - \frac{4}{3} (e^{x} + 1)^{\frac{3}{4}} + C

Graph

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