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Beliebt Rechnen >

integral of (5x^2-x)/(2sqrt(x))

Lösung

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

Lösung

\frac{1}{3} x^{\frac{3}{2}} (3 x - 1) + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Multipliziere aus

\frac{5 x ^{2} - x}{x} : \frac{5 x ^{2}}{x} - \frac{x}{x}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} x^{\frac{3}{2}} (3 x - 1) + C

Graph

Beliebte Beispiele

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

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\int x^{3} (x^{2} + 1)^{11} d x

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

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\int x^{4} x^{5} + 7 d x

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\int \frac{sin ( ln ( 11 x ) )}{x} d x