integral of (e^{2x})/(e^{2x)+3e^x+2}

Lösung

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

- ln ∣ e^{x} + 1 ∣ + 2 ln ∣ e^{x} + 2 ∣ + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Ermittle den Partialbruch von

\frac{u}{u ^{2} + 3 u + 2} : - \frac{1}{u + 1} + \frac{2}{u + 2}

= \int - \frac{1}{u + 1} + \frac{2}{u + 2} 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 \frac{1}{u + 1} d u + \int \frac{2}{u + 2} d u

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

\int \frac{2}{u + 2} d u = 2 ln ∣ u + 2 ∣

= - ln ∣ u + 1 ∣ + 2 ln ∣ u + 2 ∣

Setze in

u = e^{x}

ein

= - ln ∣ e^{x} + 1 ∣ + 2 ln ∣ e^{x} + 2 ∣

Füge eine Konstante zur Lösung hinzu

= - ln ∣ e^{x} + 1 ∣ + 2 ln ∣ e^{x} + 2 ∣ + C

\frac{\partial}{\partial x} (\frac{e ^{x y}}{x + y})

\frac{d}{d x} (3 \frac{8 x ^{2} - 3}{x ^{2} + 2})

\int \frac{e ^{3}}{x} d x

x \to 0 lim (\frac{∣ x ∣}{4 x})

\frac{d ^{2} y}{d x ^{2}} - 12 \frac{d y}{d x} + 36 y = 0