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Beliebt Rechnen >

integral of 5x^4(x^5-3)^7

Lösung

\int 5 x^{4} (x^{5} - 3)^{7} d x

Lösung

\frac{x ^{40}}{8} - 3 x^{35} + \frac{63 x ^{30}}{2} - 189 x^{25} + \frac{2835 x ^{20}}{4} - 1701 x^{15} + \frac{5103 x ^{10}}{2} - 2187 x^{5} + C

Schritte zur Lösung

\int 5 x^{4} (x^{5} - 3)^{7} d x

Entferne die Konstante:

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

= 5 \cdot \int x^{4} (x^{5} - 3)^{7} d x

Multipliziere aus

x^{4} (x^{5} - 3)^{7} : x^{39} - 21 x^{34} + 189 x^{29} - 945 x^{24} + 2835 x^{19} - 5103 x^{14} + 5103 x^{9} - 2187 x^{4}

= 5 \cdot \int x^{39} - 21 x^{34} + 189 x^{29} - 945 x^{24} + 2835 x^{19} - 5103 x^{14} + 5103 x^{9} - 2187 x^{4} 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

= 5 (\int x^{39} d x - \int 21 x^{34} d x + \int 189 x^{29} d x - \int 945 x^{24} d x + \int 2835 x^{19} d x - \int 5103 x^{14} d x + \int 5103 x^{9} d x - \int 2187 x^{4} d x)

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

\int 21 x^{34} d x = \frac{3 x ^{35}}{5}

\int 189 x^{29} d x = \frac{63 x ^{30}}{10}

\int 945 x^{24} d x = \frac{189 x ^{25}}{5}

\int 2835 x^{19} d x = \frac{567 x ^{20}}{4}

\int 5103 x^{14} d x = \frac{1701 x ^{15}}{5}

\int 5103 x^{9} d x = \frac{5103 x ^{10}}{10}

\int 2187 x^{4} d x = \frac{2187 x ^{5}}{5}

= 5 (\frac{x ^{40}}{40} - \frac{3 x ^{35}}{5} + \frac{63 x ^{30}}{10} - \frac{189 x ^{25}}{5} + \frac{567 x ^{20}}{4} - \frac{1701 x ^{15}}{5} + \frac{5103 x ^{10}}{10} - \frac{2187 x ^{5}}{5})

Vereinfache

5 (\frac{x ^{40}}{40} - \frac{3 x ^{35}}{5} + \frac{63 x ^{30}}{10} - \frac{189 x ^{25}}{5} + \frac{567 x ^{20}}{4} - \frac{1701 x ^{15}}{5} + \frac{5103 x ^{10}}{10} - \frac{2187 x ^{5}}{5}) : \frac{x ^{40}}{8} - 3 x^{35} + \frac{63 x ^{30}}{2} - 189 x^{25} + \frac{2835 x ^{20}}{4} - 1701 x^{15} + \frac{5103 x ^{10}}{2} - 2187 x^{5}

= \frac{x ^{40}}{8} - 3 x^{35} + \frac{63 x ^{30}}{2} - 189 x^{25} + \frac{2835 x ^{20}}{4} - 1701 x^{15} + \frac{5103 x ^{10}}{2} - 2187 x^{5}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{40}}{8} - 3 x^{35} + \frac{63 x ^{30}}{2} - 189 x^{25} + \frac{2835 x ^{20}}{4} - 1701 x^{15} + \frac{5103 x ^{10}}{2} - 2187 x^{5} + C

Graph

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