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

Lösung

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

2 x^{\frac{5}{2}} - \frac{5 x ^{4}}{4} + C

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Multipliziere aus

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

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

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

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

\int x^{3} d x = \frac{x ^{4}}{4}

= 5 (\frac{2}{5} x^{\frac{5}{2}} - \frac{x ^{4}}{4})

Vereinfache

5 (\frac{2}{5} x^{\frac{5}{2}} - \frac{x ^{4}}{4}) : 2 x^{\frac{5}{2}} - \frac{5 x ^{4}}{4}

= 2 x^{\frac{5}{2}} - \frac{5 x ^{4}}{4}

Füge eine Konstante zur Lösung hinzu

= 2 x^{\frac{5}{2}} - \frac{5 x ^{4}}{4} + C

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