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Beliebt Rechnen >

integral of x^7sqrt(9-x^4)

Lösung

\int x^{7} 9 - x^{4} d x

Lösung

- \frac{1}{10} (x^{4} + 6) (- x^{4} + 9)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x^{7} 9 - x^{4} d x

Wende U-Substitution an

= \int - \frac{( - u + 9 ) u}{4} d u

Entferne die Konstante:

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

= - \frac{1}{4} \cdot \int (- u + 9) u d u

Multipliziere aus

(- u + 9) u : - u^{\frac{3}{2}} + 9 u

= - \frac{1}{4} \cdot \int - u^{\frac{3}{2}} + 9 u 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}{4} (- \int u^{\frac{3}{2}} d u + \int 9 u d u)

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

\int 9 u d u = 6 u^{\frac{3}{2}}

= - \frac{1}{4} (- \frac{2}{5} u^{\frac{5}{2}} + 6 u^{\frac{3}{2}})

Setze in

u = 9 - x^{4}

ein

= - \frac{1}{4} (- \frac{2}{5} (9 - x^{4})^{\frac{5}{2}} + 6 (9 - x^{4})^{\frac{3}{2}})

Vereinfache

- \frac{1}{4} (- \frac{2}{5} (9 - x^{4})^{\frac{5}{2}} + 6 (9 - x^{4})^{\frac{3}{2}}) : - \frac{1}{10} (x^{4} + 6) (- x^{4} + 9)^{\frac{3}{2}}

= - \frac{1}{10} (x^{4} + 6) (- x^{4} + 9)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{10} (x^{4} + 6) (- x^{4} + 9)^{\frac{3}{2}} + C

Graph

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