integral of 3/(e^x+2)

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

= 3 \cdot \int \frac{1}{e ^{x} + 2} d x

Wende U-Substitution an

= 3 \cdot \int \frac{1}{u ( u + 2 )} d u

Ermittle den Partialbruch von

\frac{1}{u ( u + 2 )} : \frac{1}{2 u} - \frac{1}{2 ( u + 2 )}

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

= 3 (\int \frac{1}{2 u} d u - \int \frac{1}{2 ( u + 2 )} d u)

\int \frac{1}{2 u} d u = \frac{1}{2} ln ∣ u ∣

\int \frac{1}{2 ( u + 2 )} d u = \frac{1}{2} ln ∣ u + 2 ∣

= 3 (\frac{1}{2} ln ∣ u ∣ - \frac{1}{2} ln ∣ u + 2 ∣)

Setze in

u = e^{x}

ein

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

\frac{1}{2} ln ∣ e^{x} ∣ = \frac{1}{2} x

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (3^{5 x})

\frac{d}{d x} (9 sec (x - \frac{π}{2}))

d er i v a t i v e x^{10}

\int \frac{x - 16}{x ^{2} + 2 x - 8} d x

\frac{d}{d x} (e^{x^{2}} + 2)