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Beliebt Rechnen >

integral of x^3(4-x^2)^{1/4}

Lösung

\int x^{3} (4 - x^{2})^{\frac{1}{4}} d x

Lösung

- \frac{2}{45} (5 x^{2} + 16) (- x^{2} + 4)^{\frac{5}{4}} + C

Schritte zur Lösung

\int x^{3} (4 - x^{2})^{\frac{1}{4}} d x

Wende U-Substitution an

= \int - \frac{( - u + 4 ) u ^{\frac{1}{4}}}{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 + 4) u^{\frac{1}{4}} d u

Multipliziere aus

(- u + 4) u^{\frac{1}{4}} : - u^{\frac{5}{4}} + 4 u^{\frac{1}{4}}

= - \frac{1}{2} \cdot \int - u^{\frac{5}{4}} + 4 u^{\frac{1}{4}} 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{5}{4}} d u + \int 4 u^{\frac{1}{4}} d u)

\int u^{\frac{5}{4}} d u = \frac{4}{9} u^{\frac{9}{4}}

\int 4 u^{\frac{1}{4}} d u = \frac{16}{5} u^{\frac{5}{4}}

= - \frac{1}{2} (- \frac{4}{9} u^{\frac{9}{4}} + \frac{16}{5} u^{\frac{5}{4}})

Setze in

u = 4 - x^{2}

ein

= - \frac{1}{2} (- \frac{4}{9} (4 - x^{2})^{\frac{9}{4}} + \frac{16}{5} (4 - x^{2})^{\frac{5}{4}})

Vereinfache

- \frac{1}{2} (- \frac{4}{9} (4 - x^{2})^{\frac{9}{4}} + \frac{16}{5} (4 - x^{2})^{\frac{5}{4}}) : - \frac{2}{45} (5 x^{2} + 16) (- x^{2} + 4)^{\frac{5}{4}}

= - \frac{2}{45} (5 x^{2} + 16) (- x^{2} + 4)^{\frac{5}{4}}

Füge eine Konstante zur Lösung hinzu

= - \frac{2}{45} (5 x^{2} + 16) (- x^{2} + 4)^{\frac{5}{4}} + C

Graph

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