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Beliebt Rechnen >

integral of 6x^3(-3x^4+3)^6

Lösung

\int 6 x^{3} (- 3 x^{4} + 3)^{6} d x

Lösung

6 (\frac{729 x ^{28}}{28} - \frac{729 x ^{24}}{4} + \frac{2187 x ^{20}}{4} - \frac{3645 x ^{16}}{4} + \frac{3645 x ^{12}}{4} - \frac{2187 x ^{8}}{4} + \frac{729 x ^{4}}{4}) + C

Schritte zur Lösung

\int 6 x^{3} (- 3 x^{4} + 3)^{6} d x

Entferne die Konstante:

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

= 6 \cdot \int x^{3} (- 3 x^{4} + 3)^{6} d x

Multipliziere aus

x^{3} (- 3 x^{4} + 3)^{6} : 729 x^{27} - 4374 x^{23} + 10935 x^{19} - 14580 x^{15} + 10935 x^{11} - 4374 x^{7} + 729 x^{3}

= 6 \cdot \int 729 x^{27} - 4374 x^{23} + 10935 x^{19} - 14580 x^{15} + 10935 x^{11} - 4374 x^{7} + 729 x^{3} 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

= 6 (\int 729 x^{27} d x - \int 4374 x^{23} d x + \int 10935 x^{19} d x - \int 14580 x^{15} d x + \int 10935 x^{11} d x - \int 4374 x^{7} d x + \int 729 x^{3} d x)

\int 729 x^{27} d x = \frac{729 x ^{28}}{28}

\int 4374 x^{23} d x = \frac{729 x ^{24}}{4}

\int 10935 x^{19} d x = \frac{2187 x ^{20}}{4}

\int 14580 x^{15} d x = \frac{3645 x ^{16}}{4}

\int 10935 x^{11} d x = \frac{3645 x ^{12}}{4}

\int 4374 x^{7} d x = \frac{2187 x ^{8}}{4}

\int 729 x^{3} d x = \frac{729 x ^{4}}{4}

= 6 (\frac{729 x ^{28}}{28} - \frac{729 x ^{24}}{4} + \frac{2187 x ^{20}}{4} - \frac{3645 x ^{16}}{4} + \frac{3645 x ^{12}}{4} - \frac{2187 x ^{8}}{4} + \frac{729 x ^{4}}{4})

Füge eine Konstante zur Lösung hinzu

= 6 (\frac{729 x ^{28}}{28} - \frac{729 x ^{24}}{4} + \frac{2187 x ^{20}}{4} - \frac{3645 x ^{16}}{4} + \frac{3645 x ^{12}}{4} - \frac{2187 x ^{8}}{4} + \frac{729 x ^{4}}{4}) + C

Graph

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