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Beliebt Rechnen >

integral of x^3(x^2-5)^{1/2}

Lösung

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

Lösung

\frac{1}{15} (3 x^{2} + 10) (x^{2} - 5)^{\frac{3}{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{( u + 5 ) u ^{\frac{1}{2}}}{2} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int (u + 5) u^{\frac{1}{2}} d u

Multipliziere aus

(u + 5) u^{\frac{1}{2}} : u^{\frac{3}{2}} + 5 u^{\frac{1}{2}}

= \frac{1}{2} \cdot \int u^{\frac{3}{2}} + 5 u^{\frac{1}{2}} d u

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 u^{\frac{3}{2}} d u + \int 5 u^{\frac{1}{2}} d u)

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

\int 5 u^{\frac{1}{2}} d u = \frac{10}{3} u^{\frac{3}{2}}

= \frac{1}{2} (\frac{2}{5} u^{\frac{5}{2}} + \frac{10}{3} u^{\frac{3}{2}})

Setze in

u = x^{2} - 5

ein

= \frac{1}{2} (\frac{2}{5} (x^{2} - 5)^{\frac{5}{2}} + \frac{10}{3} (x^{2} - 5)^{\frac{3}{2}})

Vereinfache

\frac{1}{2} (\frac{2}{5} (x^{2} - 5)^{\frac{5}{2}} + \frac{10}{3} (x^{2} - 5)^{\frac{3}{2}}) : \frac{1}{15} (3 x^{2} + 10) (x^{2} - 5)^{\frac{3}{2}}

= \frac{1}{15} (3 x^{2} + 10) (x^{2} - 5)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{15} (3 x^{2} + 10) (x^{2} - 5)^{\frac{3}{2}} + C

Graph

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