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Beliebt Rechnen >

integral of x^3sqrt(4x^2-9)

Lösung

\int x^{3} 4 x^{2} - 9 d x

Lösung

\frac{1}{40} (2 x^{2} + 3) (4 x^{2} - 9)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x^{3} 4 x^{2} - 9 d x

Wende U-Substitution an

= \int \frac{u ( u + 9 )}{32} d u

Entferne die Konstante:

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

= \frac{1}{32} \cdot \int u (u + 9) d u

Multipliziere aus

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

= \frac{1}{32} \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}{32} (\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}{32} (\frac{2}{5} u^{\frac{5}{2}} + 6 u^{\frac{3}{2}})

Setze in

u = 4 x^{2} - 9

ein

= \frac{1}{32} (\frac{2}{5} (4 x^{2} - 9)^{\frac{5}{2}} + 6 (4 x^{2} - 9)^{\frac{3}{2}})

Vereinfache

\frac{1}{32} (\frac{2}{5} (4 x^{2} - 9)^{\frac{5}{2}} + 6 (4 x^{2} - 9)^{\frac{3}{2}}) : \frac{1}{40} (2 x^{2} + 3) (4 x^{2} - 9)^{\frac{3}{2}}

= \frac{1}{40} (2 x^{2} + 3) (4 x^{2} - 9)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{40} (2 x^{2} + 3) (4 x^{2} - 9)^{\frac{3}{2}} + C

Graph

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