integral of x(x+1)^8

Lösung

\int x (x + 1)^{8} d x

\frac{x ^{10}}{10} + \frac{8 x ^{9}}{9} + \frac{7 x ^{8}}{2} + 8 x^{7} + \frac{35 x ^{6}}{3} + \frac{56 x ^{5}}{5} + 7 x^{4} + \frac{8 x ^{3}}{3} + \frac{x ^{2}}{2} + C

Schritte zur Lösung

\int x (x + 1)^{8} d x

Multipliziere aus

x (x + 1)^{8} : x^{9} + 8 x^{8} + 28 x^{7} + 56 x^{6} + 70 x^{5} + 56 x^{4} + 28 x^{3} + 8 x^{2} + x

= \int x^{9} + 8 x^{8} + 28 x^{7} + 56 x^{6} + 70 x^{5} + 56 x^{4} + 28 x^{3} + 8 x^{2} + x 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^{9} d x + \int 8 x^{8} d x + \int 28 x^{7} d x + \int 56 x^{6} d x + \int 70 x^{5} d x + \int 56 x^{4} d x + \int 28 x^{3} d x + \int 8 x^{2} d x + \int x d x

\int x^{9} d x = \frac{x ^{10}}{10}

\int 8 x^{8} d x = \frac{8 x ^{9}}{9}

\int 28 x^{7} d x = \frac{7 x ^{8}}{2}

\int 56 x^{6} d x = 8 x^{7}

\int 70 x^{5} d x = \frac{35 x ^{6}}{3}

\int 56 x^{4} d x = \frac{56 x ^{5}}{5}

\int 28 x^{3} d x = 7 x^{4}

\int 8 x^{2} d x = \frac{8 x ^{3}}{3}

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

= \frac{x ^{10}}{10} + \frac{8 x ^{9}}{9} + \frac{7 x ^{8}}{2} + 8 x^{7} + \frac{35 x ^{6}}{3} + \frac{56 x ^{5}}{5} + 7 x^{4} + \frac{8 x ^{3}}{3} + \frac{x ^{2}}{2}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{10}}{10} + \frac{8 x ^{9}}{9} + \frac{7 x ^{8}}{2} + 8 x^{7} + \frac{35 x ^{6}}{3} + \frac{56 x ^{5}}{5} + 7 x^{4} + \frac{8 x ^{3}}{3} + \frac{x ^{2}}{2} + C

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\int_{- 2}^{2} (x + 5) d x

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x \to 2 lim (2)

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