integral of 6(x+2)In(x+2)

Lösung

\int 6 (x + 2) I n (x + 2) d x

6 I n (4 x + 2 x^{2} + \frac{x ^{3}}{3}) + C

Schritte zur Lösung

\int 6 (x + 2) I n (x + 2) d x

Entferne die Konstante:

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

= 6 I n \cdot \int (2 + x) (2 + x) d x

Vereinfache

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

= 6 I n \cdot \int (2 + x)^{2} d x

Multipliziere aus

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

= 6 I n \cdot \int 4 + 4 x + 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

= 6 I n (\int 4 d x + \int 4 x d x + \int x^{2} d x)

\int 4 d x = 4 x

\int 4 x d x = 2 x^{2}

\int x^{2} d x = \frac{x ^{3}}{3}

= 6 I n (4 x + 2 x^{2} + \frac{x ^{3}}{3})

Füge eine Konstante zur Lösung hinzu

= 6 I n (4 x + 2 x^{2} + \frac{x ^{3}}{3}) + C

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