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Beliebt Rechnen >

integral of (2x+1)(x^2+x+1)^5

Lösung

\int (2 x + 1) (x^{2} + x + 1)^{5} d x

Lösung

\frac{x ^{12}}{6} + x^{11} + \frac{7 x ^{10}}{2} + \frac{25 x ^{9}}{3} + 15 x^{8} + 21 x^{7} + \frac{47 x ^{6}}{2} + 21 x^{5} + 15 x^{4} + \frac{25 x ^{3}}{3} + \frac{7 x ^{2}}{2} + x + C

Schritte zur Lösung

\int (2 x + 1) (x^{2} + x + 1)^{5} d x

Multipliziere aus

(2 x + 1) (x^{2} + x + 1)^{5} : 2 x^{11} + 11 x^{10} + 35 x^{9} + 75 x^{8} + 120 x^{7} + 147 x^{6} + 141 x^{5} + 105 x^{4} + 60 x^{3} + 25 x^{2} + 7 x + 1

= \int 2 x^{11} + 11 x^{10} + 35 x^{9} + 75 x^{8} + 120 x^{7} + 147 x^{6} + 141 x^{5} + 105 x^{4} + 60 x^{3} + 25 x^{2} + 7 x + 1 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 2 x^{11} d x + \int 11 x^{10} d x + \int 35 x^{9} d x + \int 75 x^{8} d x + \int 120 x^{7} d x + \int 147 x^{6} d x + \int 141 x^{5} d x + \int 105 x^{4} d x + \int 60 x^{3} d x + \int 25 x^{2} d x + \int 7 x d x + \int 1 d x

\int 2 x^{11} d x = \frac{x ^{12}}{6}

\int 11 x^{10} d x = x^{11}

\int 35 x^{9} d x = \frac{7 x ^{10}}{2}

\int 75 x^{8} d x = \frac{25 x ^{9}}{3}

\int 120 x^{7} d x = 15 x^{8}

\int 147 x^{6} d x = 21 x^{7}

\int 141 x^{5} d x = \frac{47 x ^{6}}{2}

\int 105 x^{4} d x = 21 x^{5}

\int 60 x^{3} d x = 15 x^{4}

\int 25 x^{2} d x = \frac{25 x ^{3}}{3}

\int 7 x d x = \frac{7 x ^{2}}{2}

\int 1 d x = x

= \frac{x ^{12}}{6} + x^{11} + \frac{7 x ^{10}}{2} + \frac{25 x ^{9}}{3} + 15 x^{8} + 21 x^{7} + \frac{47 x ^{6}}{2} + 21 x^{5} + 15 x^{4} + \frac{25 x ^{3}}{3} + \frac{7 x ^{2}}{2} + x

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{12}}{6} + x^{11} + \frac{7 x ^{10}}{2} + \frac{25 x ^{9}}{3} + 15 x^{8} + 21 x^{7} + \frac{47 x ^{6}}{2} + 21 x^{5} + 15 x^{4} + \frac{25 x ^{3}}{3} + \frac{7 x ^{2}}{2} + x + C

Graph

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