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

Lösung

integral

\frac{1}{( x ^{\frac{1}{2}} + 1 )}

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l (x^{2} - x)^{4}

in t e g r a l \frac{2}{( 2 x + 3 )}

in t e g r a l \frac{e ^{x}}{e ^{x} + 1}

in t e g r a l \frac{e ^{- x}}{( 1 + e ^{- x} )}

in t e g r a l x sin^{2} (x)