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Beliebt Rechnen >

integral of ln(x^2+2x)

Lösung

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

Lösung

x ln (x^{2} + 2 x) + ln (x^{2} + 2 x) + ln ∣ x + 2 ∣ - ln ∣ x ∣ - 2 x - 2 + C

Schritte zur Lösung

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

Vervollständige das Quadrat

x^{2} + 2 x : (x + 1)^{2} - 1

= \int ln ((x + 1)^{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 = - ln ∣ u + 1 ∣ + ln ∣ u - 1 ∣ + 2 u

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

Setze in

u = x + 1

ein

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

Vereinfache

(x + 1) ln ((x + 1)^{2} - 1) - (- ln ∣ x + 1 + 1 ∣ + ln ∣ x + 1 - 1 ∣ + 2 (x + 1)) : x ln (x^{2} + 2 x) + ln (x^{2} + 2 x) + ln ∣ x + 2 ∣ - ln ∣ x ∣ - 2 x - 2

= x ln (x^{2} + 2 x) + ln (x^{2} + 2 x) + ln ∣ x + 2 ∣ - ln ∣ x ∣ - 2 x - 2

Füge eine Konstante zur Lösung hinzu

= x ln (x^{2} + 2 x) + ln (x^{2} + 2 x) + ln ∣ x + 2 ∣ - ln ∣ x ∣ - 2 x - 2 + C

Graph

Beliebte Beispiele

integral of (3x^2+19x+27)/((x+2)(x+3)^2)

\int \frac{3 x ^{2} + 19 x + 27}{( x + 2 ) ( x + 3 ) ^{2}} d x

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\int 10 e^{- \frac{x}{2}} d x

integral of (1-x)e^{x^2}

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

integral of 1/((sqrt(9-x^2))^3)

\int \frac{1}{( 9 - x ^{2} ) ^{3}} d x

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

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