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Beliebt Rechnen >

integral of e^{2t}arctan(e^t)

Lösung

\int e^{2 t} arctan (e^{t}) d t

Lösung

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

Schritte zur Lösung

\int e^{2 t} arctan (e^{t}) d t

Wende U-Substitution an

= \int u arctan (u) d u

Wende die partielle Integration an

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

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

= \frac{1}{2} u^{2} arctan (u) - \frac{1}{2} (- arctan (u) + u)

Setze in

u = e^{t}

ein

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

Wende Exponentenregel an:

(a^{b})^{c} = a^{b c}

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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