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Beliebt Rechnen >

integral of ((1-x)^3)/((x^3sqrt(x)))

Lösung

\int \frac{( 1 - x ) ^{3}}{( x ^{3} x )} d x

Lösung

- \frac{2}{5 x ^{\frac{5}{2}}} + \frac{2}{x ^{\frac{3}{2}}} - \frac{6}{x} - 2 x + C

Schritte zur Lösung

\int \frac{( 1 - x ) ^{3}}{x ^{3} x} d x

Multipliziere aus

\frac{( 1 - x ) ^{3}}{x ^{3} x} : \frac{1}{x ^{\frac{7}{2}}} - \frac{3}{x ^{\frac{5}{2}}} + \frac{3}{x ^{\frac{3}{2}}} - \frac{1}{x ^{\frac{1}{2}}}

= \int \frac{1}{x ^{\frac{7}{2}}} - \frac{3}{x ^{\frac{5}{2}}} + \frac{3}{x ^{\frac{3}{2}}} - \frac{1}{x ^{\frac{1}{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 \frac{1}{x ^{\frac{7}{2}}} d x - \int \frac{3}{x ^{\frac{5}{2}}} d x + \int \frac{3}{x ^{\frac{3}{2}}} d x - \int \frac{1}{x ^{\frac{1}{2}}} d x

\int \frac{1}{x ^{\frac{7}{2}}} d x = - \frac{2}{5 x ^{\frac{5}{2}}}

\int \frac{3}{x ^{\frac{5}{2}}} d x = - \frac{2}{x ^{\frac{3}{2}}}

\int \frac{3}{x ^{\frac{3}{2}}} d x = - \frac{6}{x}

\int \frac{1}{x ^{\frac{1}{2}}} d x = 2 x^{\frac{1}{2}}

= - \frac{2}{5 x ^{\frac{5}{2}}} - (- \frac{2}{x ^{\frac{3}{2}}}) - \frac{6}{x} - 2 x^{\frac{1}{2}}

Vereinfache

- \frac{2}{5 x ^{\frac{5}{2}}} - (- \frac{2}{x ^{\frac{3}{2}}}) - \frac{6}{x} - 2 x^{\frac{1}{2}} : - \frac{2}{5 x ^{\frac{5}{2}}} + \frac{2}{x ^{\frac{3}{2}}} - \frac{6}{x} - 2 x

= - \frac{2}{5 x ^{\frac{5}{2}}} + \frac{2}{x ^{\frac{3}{2}}} - \frac{6}{x} - 2 x

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{5 x ^{\frac{5}{2}}} + \frac{2}{x ^{\frac{3}{2}}} - \frac{6}{x} - 2 x + C

Graph

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