integral of e^{2x}arctan(sqrt(e^x-1))

Lösung

\int e^{2 x} arctan (e^{x} - 1) d x

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

Schritte zur Lösung

\int e^{2 x} arctan (e^{x} - 1) d x

Wende die partielle Integration an

= \frac{1}{2} e^{2 x} arctan (e^{x} - 1) - \int \frac{e ^{2 x}}{4 e ^{x} - 1} d x

\int \frac{e ^{2 x}}{4 e ^{x} - 1} d x = \frac{1}{2} (\frac{1}{3} (e^{x} - 1)^{\frac{3}{2}} + e^{x} - 1)

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

Füge eine Konstante zur Lösung hinzu

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

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