integral of 3(x+4)^2

Lösung

\int 3 (x + 4)^{2} d x

x^{3} + 12 x^{2} + 48 x + C

Schritte zur Lösung

\int 3 (x + 4)^{2} d x

Entferne die Konstante:

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

= 3 \cdot \int (x + 4)^{2} d x

Multipliziere aus

(x + 4)^{2} : x^{2} + 8 x + 16

= 3 \cdot \int x^{2} + 8 x + 16 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

= 3 (\int x^{2} d x + \int 8 x d x + \int 16 d x)

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

\int 8 x d x = 4 x^{2}

\int 16 d x = 16 x

= 3 (\frac{x ^{3}}{3} + 4 x^{2} + 16 x)

Vereinfache

3 (\frac{x ^{3}}{3} + 4 x^{2} + 16 x) : x^{3} + 12 x^{2} + 48 x

= x^{3} + 12 x^{2} + 48 x

Füge eine Konstante zur Lösung hinzu

= x^{3} + 12 x^{2} + 48 x + C