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Beliebt Rechnen >

integral of x^2ln(x^6-1)

Lösung

\int x^{2} ln (x^{6} - 1) d x

Lösung

\frac{1}{3} (x^{3} ln (x^{6} - 1) - 2 x^{3} + ln x^{3} + 1 - ln x^{3} - 1) + C

Schritte zur Lösung

\int x^{2} ln (x^{6} - 1) d x

Wende U-Substitution an

= \int \frac{ln ( u - 1 )}{6 u} d u

Entferne die Konstante:

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

= \frac{1}{6} \cdot \int \frac{ln ( u - 1 )}{u} d u

Wende die partielle Integration an

= \frac{1}{6} (2 u ln (u - 1) - \int \frac{2 u}{u - 1} d u)

\int \frac{2 u}{u - 1} d u = - 4 (\frac{1}{2} ln u + 1 - \frac{1}{2} ln u - 1 - u)

= \frac{1}{6} (2 u ln (u - 1) - (- 4 (\frac{1}{2} ln u + 1 - \frac{1}{2} ln u - 1 - u)))

Setze in

u = x^{6}

ein

= \frac{1}{6} (2 x^{6} ln (x^{6} - 1) - (- 4 (\frac{1}{2} ln x^{6} + 1 - \frac{1}{2} ln x^{6} - 1 - x^{6})))

Vereinfache

\frac{1}{6} (2 x^{6} ln (x^{6} - 1) - (- 4 (\frac{1}{2} ln x^{6} + 1 - \frac{1}{2} ln x^{6} - 1 - x^{6}))) : \frac{1}{3} (x^{3} ln (x^{6} - 1) - 2 x^{3} + ln x^{3} + 1 - ln x^{3} - 1)

= \frac{1}{3} (x^{3} ln (x^{6} - 1) - 2 x^{3} + ln x^{3} + 1 - ln x^{3} - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} (x^{3} ln (x^{6} - 1) - 2 x^{3} + ln x^{3} + 1 - ln x^{3} - 1) + C

Graph

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