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Beliebt Rechnen >

integral of x^2ln(x^3+1)

Lösung

\int x^{2} ln (x^{3} + 1) d x

Lösung

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

Schritte zur Lösung

\int x^{2} ln (x^{3} + 1) d x

Wende U-Substitution an

= \int \frac{ln ( u )}{3} d u

Entferne die Konstante:

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

= \frac{1}{3} \cdot \int ln (u) d u

Wende die partielle Integration an

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

\int 1 d u = u

= \frac{1}{3} (u ln (u) - u)

Setze in

u = x^{3} + 1

ein

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

Multipliziere aus

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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