integral of x^2(5x^3-2)^4

Lösung

\int x^{2} (5 x^{3} - 2)^{4} d x

\frac{125 x ^{15}}{3} - \frac{250 x ^{12}}{3} + \frac{200 x ^{9}}{3} - \frac{80 x ^{6}}{3} + \frac{16 x ^{3}}{3} + C

Schritte zur Lösung

\int x^{2} (5 x^{3} - 2)^{4} d x

Multipliziere aus

x^{2} (5 x^{3} - 2)^{4} : 625 x^{14} - 1000 x^{11} + 600 x^{8} - 160 x^{5} + 16 x^{2}

= \int 625 x^{14} - 1000 x^{11} + 600 x^{8} - 160 x^{5} + 16 x^{2} 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 625 x^{14} d x - \int 1000 x^{11} d x + \int 600 x^{8} d x - \int 160 x^{5} d x + \int 16 x^{2} d x

\int 625 x^{14} d x = \frac{125 x ^{15}}{3}

\int 1000 x^{11} d x = \frac{250 x ^{12}}{3}

\int 600 x^{8} d x = \frac{200 x ^{9}}{3}

\int 160 x^{5} d x = \frac{80 x ^{6}}{3}

\int 16 x^{2} d x = \frac{16 x ^{3}}{3}

= \frac{125 x ^{15}}{3} - \frac{250 x ^{12}}{3} + \frac{200 x ^{9}}{3} - \frac{80 x ^{6}}{3} + \frac{16 x ^{3}}{3}

Füge eine Konstante zur Lösung hinzu

= \frac{125 x ^{15}}{3} - \frac{250 x ^{12}}{3} + \frac{200 x ^{9}}{3} - \frac{80 x ^{6}}{3} + \frac{16 x ^{3}}{3} + C

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