integral 1/((x^{1/3)+1)}

Lösung

integral

\frac{1}{( x ^{\frac{1}{3}} + 1 )}

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{3 u ^{2}}{u + 1} d u

Entferne die Konstante:

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

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

Wende U-Substitution an

= 3 \cdot \int \frac{( v - 1 ) ^{2}}{v} d v

Multipliziere aus

\frac{( v - 1 ) ^{2}}{v} : v - 2 + \frac{1}{v}

= 3 \cdot \int v - 2 + \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

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

\int v d v = \frac{v ^{2}}{2}

\int 2 d v = 2 v

\int \frac{1}{v} d v = ln ∣ v ∣

= 3 (\frac{v ^{2}}{2} - 2 v + ln ∣ v ∣)

Ersetze zurück

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l 2 x cos (x^{2} - 5)

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

in t e g r a l \frac{( e ^{- x} ) ^{2}}{b}

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in t e g r a l \frac{1}{ln ^{2} ( x )}