integral of x^4(x^5-7)^6

Lösung

\int x^{4} (x^{5} - 7)^{6} d x

\frac{x ^{35}}{35} - \frac{7 x ^{30}}{5} + \frac{147 x ^{25}}{5} - 343 x^{20} + 2401 x^{15} - \frac{50421 x ^{10}}{5} + \frac{117649 x ^{5}}{5} + C

Schritte zur Lösung

\int x^{4} (x^{5} - 7)^{6} d x

Multipliziere aus

x^{4} (x^{5} - 7)^{6} : x^{34} - 42 x^{29} + 735 x^{24} - 6860 x^{19} + 36015 x^{14} - 100842 x^{9} + 117649 x^{4}

= \int x^{34} - 42 x^{29} + 735 x^{24} - 6860 x^{19} + 36015 x^{14} - 100842 x^{9} + 117649 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

= \int x^{34} d x - \int 42 x^{29} d x + \int 735 x^{24} d x - \int 6860 x^{19} d x + \int 36015 x^{14} d x - \int 100842 x^{9} d x + \int 117649 x^{4} d x

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

\int 42 x^{29} d x = \frac{7 x ^{30}}{5}

\int 735 x^{24} d x = \frac{147 x ^{25}}{5}

\int 6860 x^{19} d x = 343 x^{20}

\int 36015 x^{14} d x = 2401 x^{15}

\int 100842 x^{9} d x = \frac{50421 x ^{10}}{5}

\int 117649 x^{4} d x = \frac{117649 x ^{5}}{5}

= \frac{x ^{35}}{35} - \frac{7 x ^{30}}{5} + \frac{147 x ^{25}}{5} - 343 x^{20} + 2401 x^{15} - \frac{50421 x ^{10}}{5} + \frac{117649 x ^{5}}{5}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{35}}{35} - \frac{7 x ^{30}}{5} + \frac{147 x ^{25}}{5} - 343 x^{20} + 2401 x^{15} - \frac{50421 x ^{10}}{5} + \frac{117649 x ^{5}}{5} + C

in v erse l a pl a ce \frac{8}{s - 3}

\int 3 t^{5} d t

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