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Beliebt Rechnen >

integral of x^4(1-x^5)^7

Lösung

\int x^{4} (1 - x^{5})^{7} d x

Lösung

\frac{x ^{5}}{5} - \frac{7 x ^{10}}{10} + \frac{7 x ^{15}}{5} - \frac{7 x ^{20}}{4} + \frac{7 x ^{25}}{5} - \frac{7 x ^{30}}{10} + \frac{x ^{35}}{5} - \frac{x ^{40}}{40} + C

Schritte zur Lösung

\int x^{4} (1 - x^{5})^{7} d x

Multipliziere aus

x^{4} (1 - x^{5})^{7} : x^{4} - 7 x^{9} + 21 x^{14} - 35 x^{19} + 35 x^{24} - 21 x^{29} + 7 x^{34} - x^{39}

= \int x^{4} - 7 x^{9} + 21 x^{14} - 35 x^{19} + 35 x^{24} - 21 x^{29} + 7 x^{34} - x^{39} 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^{4} d x - \int 7 x^{9} d x + \int 21 x^{14} d x - \int 35 x^{19} d x + \int 35 x^{24} d x - \int 21 x^{29} d x + \int 7 x^{34} d x - \int x^{39} d x

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

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

\int 21 x^{14} d x = \frac{7 x ^{15}}{5}

\int 35 x^{19} d x = \frac{7 x ^{20}}{4}

\int 35 x^{24} d x = \frac{7 x ^{25}}{5}

\int 21 x^{29} d x = \frac{7 x ^{30}}{10}

\int 7 x^{34} d x = \frac{x ^{35}}{5}

\int x^{39} d x = \frac{x ^{40}}{40}

= \frac{x ^{5}}{5} - \frac{7 x ^{10}}{10} + \frac{7 x ^{15}}{5} - \frac{7 x ^{20}}{4} + \frac{7 x ^{25}}{5} - \frac{7 x ^{30}}{10} + \frac{x ^{35}}{5} - \frac{x ^{40}}{40}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{5}}{5} - \frac{7 x ^{10}}{10} + \frac{7 x ^{15}}{5} - \frac{7 x ^{20}}{4} + \frac{7 x ^{25}}{5} - \frac{7 x ^{30}}{10} + \frac{x ^{35}}{5} - \frac{x ^{40}}{40} + C

Graph

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