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Beliebt Rechnen >

integral of ((1+e^x)^2)/(1+e^{2x)}

Lösung

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

Lösung

2 arctan (e^{x}) + x + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

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

= \frac{1}{2} \cdot \int \frac{2 e ^{\frac{u}{2}}}{1 + e ^{u}} + 1 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

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

\int \frac{2 e ^{\frac{u}{2}}}{1 + e ^{u}} d u = 4 arctan (e^{\frac{u}{2}})

\int 1 d u = u

= \frac{1}{2} (4 arctan (e^{\frac{u}{2}}) + u)

Setze in

u = 2 x

ein

= \frac{1}{2} (4 arctan (e^{\frac{2 x}{2}}) + 2 x)

Vereinfache

\frac{1}{2} (4 arctan (e^{\frac{2 x}{2}}) + 2 x) : 2 arctan (e^{x}) + x

= 2 arctan (e^{x}) + x

Füge eine Konstante zur Lösung hinzu

= 2 arctan (e^{x}) + x + C

Graph

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