integral of ln(x^2+4x+5)

Lösung

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

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

Schritte zur Lösung

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

Vervollständige das Quadrat

x^{2} + 4 x + 5 : (x + 2)^{2} + 1

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

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 = x + 2

ein

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

Multipliziere aus

(x + 2)^{2} + 1 : x^{2} + 4 x + 5

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

Füge eine Konstante zur Lösung hinzu

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

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