integral of (x^2)/(sqrt(x^3)-1)

Lösung

\int \frac{x ^{2}}{x ^{3} - 1} d x

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

Schritte zur Lösung

\int \frac{x ^{2}}{x ^{3} - 1} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Multipliziere aus

\frac{v + 1}{v} : 1 + \frac{1}{v}

= \frac{2}{3} \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

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

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= \frac{2}{3} (v + ln ∣ v ∣)

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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