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Beliebt Rechnen >

integral of ln(t^2-4t+5)

Lösung

\int ln (t^{2} - 4 t + 5) d t

Lösung

ln (t^{2} - 4 t + 5) (t - 2) - 2 (- arctan (t - 2) + t - 2) + C

Schritte zur Lösung

\int ln (t^{2} - 4 t + 5) d t

Vervollständige das Quadrat

t^{2} - 4 t + 5 : (t - 2)^{2} + 1

= \int ln ((t - 2)^{2} + 1) d t

Wende U-Substitution an

= \int ln (u^{2} + 1) d u

Wende die partielle Integration an

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

\int \frac{2 u ^{2}}{u ^{2} + 1} d u = 2 (- arctan (u) + u)

= u ln (u^{2} + 1) - 2 (- arctan (u) + u)

Setze in

u = t - 2

ein

= (t - 2) ln ((t - 2)^{2} + 1) - 2 (- arctan (t - 2) + t - 2)

Multipliziere aus

(t - 2)^{2} + 1 : t^{2} - 4 t + 5

= ln (t^{2} - 4 t + 5) (t - 2) - 2 (- arctan (t - 2) + t - 2)

Füge eine Konstante zur Lösung hinzu

= ln (t^{2} - 4 t + 5) (t - 2) - 2 (- arctan (t - 2) + t - 2) + C

Graph

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