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Beliebt Rechnen >

integral of 1/((1+\sqrt[3]{x))}

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

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\int csc (\frac{5}{x}) (\frac{1}{x ^{2}}) d x

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\int \frac{( 2 x - 3 )}{( x ^{2} - 3 x + 6 )} d x

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\int (2 x^{3} + x) arctan (x) d x