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Beliebt Rechnen >

integral of (1+e^{-2y})^{3/2}(e^{-2y})

Lösung

\int (1 + e^{- 2 y})^{\frac{3}{2}} (e^{- 2 y}) d y

Lösung

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

Schritte zur Lösung

\int (1 + e^{- 2 y})^{\frac{3}{2}} e^{- 2 y} d y

Wende U-Substitution an

= \int - \frac{u ^{\frac{3}{2}}}{2} d u

Entferne die Konstante:

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

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

Wende die Potenzregel an

= - \frac{1}{2} \cdot \frac{2}{5} u^{\frac{5}{2}}

Setze in

u = 1 + e^{- 2 y}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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