integral of (x^7+2x)^5

Lösung

\int (x^{7} + 2 x)^{5} d x

\frac{x ^{36}}{36} + \frac{x ^{30}}{3} + \frac{5 x ^{24}}{3} + \frac{40 x ^{18}}{9} + \frac{20 x ^{12}}{3} + \frac{16 x ^{6}}{3} + C

Schritte zur Lösung

\int (x^{7} + 2 x)^{5} d x

Multipliziere aus

(x^{7} + 2 x)^{5} : x^{35} + 10 x^{29} + 40 x^{23} + 80 x^{17} + 80 x^{11} + 32 x^{5}

= \int x^{35} + 10 x^{29} + 40 x^{23} + 80 x^{17} + 80 x^{11} + 32 x^{5} 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

= \int x^{35} d x + \int 10 x^{29} d x + \int 40 x^{23} d x + \int 80 x^{17} d x + \int 80 x^{11} d x + \int 32 x^{5} d x

\int x^{35} d x = \frac{x ^{36}}{36}

\int 10 x^{29} d x = \frac{x ^{30}}{3}

\int 40 x^{23} d x = \frac{5 x ^{24}}{3}

\int 80 x^{17} d x = \frac{40 x ^{18}}{9}

\int 80 x^{11} d x = \frac{20 x ^{12}}{3}

\int 32 x^{5} d x = \frac{16 x ^{6}}{3}

= \frac{x ^{36}}{36} + \frac{x ^{30}}{3} + \frac{5 x ^{24}}{3} + \frac{40 x ^{18}}{9} + \frac{20 x ^{12}}{3} + \frac{16 x ^{6}}{3}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{36}}{36} + \frac{x ^{30}}{3} + \frac{5 x ^{24}}{3} + \frac{40 x ^{18}}{9} + \frac{20 x ^{12}}{3} + \frac{16 x ^{6}}{3} + C