integral of (e^x)/(1+sqrt(e^x))

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Wende U-Substitution an

= \int \frac{2 ( v - 1 )}{v} d v

Entferne die Konstante:

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

= 2 \cdot \int \frac{v - 1}{v} d v

Multipliziere aus

\frac{v - 1}{v} : 1 - \frac{1}{v}

= 2 \cdot \int 1 - \frac{1}{v} d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int 1 d v - \int \frac{1}{v} d v)

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= 2 (v - ln ∣ v ∣)

Ersetze zurück

= 2 (1 + e^{x} - ln 1 + e^{x})

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

d er i v a t i v e y = \frac{x ^{2}}{x - 4}

d er i v a t i v e f (x) = 1 + x^{2}

\int x (x + 2)^{4} d x

t an g e n t f (x) = 4 x^{3} - 12 x^{2}, at x = - 2

\int sin (9 t) d t