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Beliebt Rechnen >

integral of (ln(x))^6

Lösung

\int (ln (x))^{6} d x

Lösung

x ln^{6} (x) - 6 (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))) + C

Schritte zur Lösung

\int (ln (x))^{6} d x

Vereinfache

= \int ln^{6} (x) d x

Wende U-Substitution an

= \int u^{6} e^{u} d u

Wende die partielle Integration an

= u^{6} e^{u} - \int 6 u^{5} e^{u} d u

\int 6 u^{5} e^{u} d u = 6 (u^{5} e^{u} - 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))))

= u^{6} e^{u} - 6 (u^{5} e^{u} - 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))))

Setze in

u = ln (x)

ein

= ln^{6} (x) e^{l n (x)} - 6 (ln^{5} (x) e^{l n (x)} - 5 (ln^{4} (x) e^{l n (x)} - 4 (ln^{3} (x) e^{l n (x)} - 3 (ln^{2} (x) e^{l n (x)} - 2 (e^{l n (x)} ln (x) - e^{l n (x)})))))

Vereinfache

ln^{6} (x) e^{l n (x)} - 6 (ln^{5} (x) e^{l n (x)} - 5 (ln^{4} (x) e^{l n (x)} - 4 (ln^{3} (x) e^{l n (x)} - 3 (ln^{2} (x) e^{l n (x)} - 2 (e^{l n (x)} ln (x) - e^{l n (x)}))))) : x ln^{6} (x) - 6 (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)))))

= x ln^{6} (x) - 6 (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)))))

Füge eine Konstante zur Lösung hinzu

= x ln^{6} (x) - 6 (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))) + C

Graph

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