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Beliebt Rechnen >

integral of ((x^2)/4-x/2)^5(x-1)

Lösung

\int (\frac{x ^{2}}{4} - \frac{x}{2})^{5} (x - 1) d x

Lösung

\frac{x ^{12}}{12288} - \frac{x ^{11}}{1024} + \frac{5 x ^{10}}{1024} - \frac{5 x ^{9}}{384} + \frac{5 x ^{8}}{256} - \frac{x ^{7}}{64} + \frac{x ^{6}}{192} + C

Schritte zur Lösung

\int (\frac{x ^{2}}{4} - \frac{x}{2})^{5} (x - 1) d x

Multipliziere aus

(\frac{x ^{2}}{4} - \frac{x}{2})^{5} (x - 1) : \frac{x ^{11}}{1024} - \frac{11 x ^{10}}{1024} + \frac{25 x ^{9}}{512} - \frac{15 x ^{8}}{128} + \frac{5 x ^{7}}{32} - \frac{7 x ^{6}}{64} + \frac{x ^{5}}{32}

= \int \frac{x ^{11}}{1024} - \frac{11 x ^{10}}{1024} + \frac{25 x ^{9}}{512} - \frac{15 x ^{8}}{128} + \frac{5 x ^{7}}{32} - \frac{7 x ^{6}}{64} + \frac{x ^{5}}{32} 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 \frac{x ^{11}}{1024} d x - \int \frac{11 x ^{10}}{1024} d x + \int \frac{25 x ^{9}}{512} d x - \int \frac{15 x ^{8}}{128} d x + \int \frac{5 x ^{7}}{32} d x - \int \frac{7 x ^{6}}{64} d x + \int \frac{x ^{5}}{32} d x

\int \frac{x ^{11}}{1024} d x = \frac{x ^{12}}{12288}

\int \frac{11 x ^{10}}{1024} d x = \frac{x ^{11}}{1024}

\int \frac{25 x ^{9}}{512} d x = \frac{5 x ^{10}}{1024}

\int \frac{15 x ^{8}}{128} d x = \frac{5 x ^{9}}{384}

\int \frac{5 x ^{7}}{32} d x = \frac{5 x ^{8}}{256}

\int \frac{7 x ^{6}}{64} d x = \frac{x ^{7}}{64}

\int \frac{x ^{5}}{32} d x = \frac{x ^{6}}{192}

= \frac{x ^{12}}{12288} - \frac{x ^{11}}{1024} + \frac{5 x ^{10}}{1024} - \frac{5 x ^{9}}{384} + \frac{5 x ^{8}}{256} - \frac{x ^{7}}{64} + \frac{x ^{6}}{192}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{12}}{12288} - \frac{x ^{11}}{1024} + \frac{5 x ^{10}}{1024} - \frac{5 x ^{9}}{384} + \frac{5 x ^{8}}{256} - \frac{x ^{7}}{64} + \frac{x ^{6}}{192} + C

Graph

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