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Beliebt Rechnen >

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

Lösung

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

Lösung

x^{2} ln (x + 2) - \frac{1}{2} x^{2} + 4 x ln (x + 2) - 2 x + 4 ln (x + 2) - 2 + C

Schritte zur Lösung

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

Vereinfache

ln ((x + 2)^{2}) : 2 ln (x + 2)

= \int 2 ln (x + 2) (x + 2) d x

Entferne die Konstante:

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

= 2 \cdot \int ln (x + 2) (x + 2) d x

Multipliziere aus

ln (x + 2) (x + 2) : x ln (x + 2) + 2 ln (x + 2)

= 2 \cdot \int x ln (x + 2) + 2 ln (x + 2) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int x ln (x + 2) d x + \int 2 ln (x + 2) d x)

\int x ln (x + 2) d x = ln (x + 2) (\frac{1}{2} (x + 2)^{2} - 2 (x + 2)) - \frac{1}{4} (x^{2} - 4 x - 12)

\int 2 ln (x + 2) d x = 2 (x ln (x + 2) + 2 ln (x + 2) - x - 2)

= 2 (ln (x + 2) (\frac{1}{2} (x + 2)^{2} - 2 (x + 2)) - \frac{1}{4} (x^{2} - 4 x - 12) + 2 (x ln (x + 2) + 2 ln (x + 2) - x - 2))

Vereinfache

2 (ln (x + 2) (\frac{1}{2} (x + 2)^{2} - 2 (x + 2)) - \frac{1}{4} (x^{2} - 4 x - 12) + 2 (x ln (x + 2) + 2 ln (x + 2) - x - 2)) : x^{2} ln (x + 2) - \frac{1}{2} x^{2} + 4 x ln (x + 2) - 2 x + 4 ln (x + 2) - 2

= x^{2} ln (x + 2) - \frac{1}{2} x^{2} + 4 x ln (x + 2) - 2 x + 4 ln (x + 2) - 2

Füge eine Konstante zur Lösung hinzu

= x^{2} ln (x + 2) - \frac{1}{2} x^{2} + 4 x ln (x + 2) - 2 x + 4 ln (x + 2) - 2 + C

Graph

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\frac{d}{d x} (x \cdot ln (x^{2}))

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