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Beliebt Rechnen >

integral of (e^2x)/((1+3e^2x))

Lösung

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

Lösung

\frac{1}{9 e ^{2}} + \frac{1}{3} x - \frac{1}{9 e ^{2}} ln 1 + 3 e^{2} x + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende U-Substitution an

= e^{2} \cdot \int \frac{u - 1}{9 e ^{4} u} d u

Entferne die Konstante:

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

= e^{2} \frac{1}{9 e ^{4}} \cdot \int \frac{u - 1}{u} d u

Multipliziere aus

\frac{u - 1}{u} : 1 - \frac{1}{u}

= e^{2} \frac{1}{9 e ^{4}} \cdot \int 1 - \frac{1}{u} 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

= e^{2} \frac{1}{9 e ^{4}} (\int 1 d u - \int \frac{1}{u} d u)

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= e^{2} \frac{1}{9 e ^{4}} (u - ln ∣ u ∣)

Setze in

u = 1 + 3 e^{2} x

ein

= e^{2} \frac{1}{9 e ^{4}} (1 + 3 e^{2} x - ln 1 + 3 e^{2} x)

Vereinfache

e^{2} \frac{1}{9 e ^{4}} (1 + 3 e^{2} x - ln 1 + 3 e^{2} x) : \frac{1}{9 e ^{2}} + \frac{1}{3} x - \frac{1}{9 e ^{2}} ln 1 + 3 e^{2} x

= \frac{1}{9 e ^{2}} + \frac{1}{3} x - \frac{1}{9 e ^{2}} ln 1 + 3 e^{2} x

Füge eine Konstante zur Lösung hinzu

= \frac{1}{9 e ^{2}} + \frac{1}{3} x - \frac{1}{9 e ^{2}} ln 1 + 3 e^{2} x + C

Graph

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