integral of 30(x+2)(x+3)^4

Lösung

\int 30 (x + 2) (x + 3)^{4} d x

30 (\frac{x ^{6}}{6} + \frac{14 x ^{5}}{5} + \frac{39 x ^{4}}{2} + 72 x^{3} + \frac{297 x ^{2}}{2} + 162 x) + C

Schritte zur Lösung

\int 30 (x + 2) (x + 3)^{4} d x

Entferne die Konstante:

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

= 30 \cdot \int (x + 2) (x + 3)^{4} d x

Multipliziere aus

(x + 2) (x + 3)^{4} : x^{5} + 14 x^{4} + 78 x^{3} + 216 x^{2} + 297 x + 162

= 30 \cdot \int x^{5} + 14 x^{4} + 78 x^{3} + 216 x^{2} + 297 x + 162 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

= 30 (\int x^{5} d x + \int 14 x^{4} d x + \int 78 x^{3} d x + \int 216 x^{2} d x + \int 297 x d x + \int 162 d x)

\int x^{5} d x = \frac{x ^{6}}{6}

\int 14 x^{4} d x = \frac{14 x ^{5}}{5}

\int 78 x^{3} d x = \frac{39 x ^{4}}{2}

\int 216 x^{2} d x = 72 x^{3}

\int 297 x d x = \frac{297 x ^{2}}{2}

\int 162 d x = 162 x

= 30 (\frac{x ^{6}}{6} + \frac{14 x ^{5}}{5} + \frac{39 x ^{4}}{2} + 72 x^{3} + \frac{297 x ^{2}}{2} + 162 x)

Füge eine Konstante zur Lösung hinzu

= 30 (\frac{x ^{6}}{6} + \frac{14 x ^{5}}{5} + \frac{39 x ^{4}}{2} + 72 x^{3} + \frac{297 x ^{2}}{2} + 162 x) + C