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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{63} (7 x^{2} - 4) (x^{2} + 2)^{\frac{7}{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

(u - 2) u^{\frac{5}{2}} : u^{\frac{7}{2}} - 2 u^{\frac{5}{2}}

= \frac{1}{2} \cdot \int u^{\frac{7}{2}} - 2 u^{\frac{5}{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{7}{2}} d u - \int 2 u^{\frac{5}{2}} d u)

\int u^{\frac{7}{2}} d u = \frac{2}{9} u^{\frac{9}{2}}

\int 2 u^{\frac{5}{2}} d u = \frac{4}{7} u^{\frac{7}{2}}

= \frac{1}{2} (\frac{2}{9} u^{\frac{9}{2}} - \frac{4}{7} u^{\frac{7}{2}})

Setze in

u = 2 + x^{2}

ein

= \frac{1}{2} (\frac{2}{9} (2 + x^{2})^{\frac{9}{2}} - \frac{4}{7} (2 + x^{2})^{\frac{7}{2}})

Vereinfache

\frac{1}{2} (\frac{2}{9} (2 + x^{2})^{\frac{9}{2}} - \frac{4}{7} (2 + x^{2})^{\frac{7}{2}}) : \frac{1}{63} (7 x^{2} - 4) (x^{2} + 2)^{\frac{7}{2}}

= \frac{1}{63} (7 x^{2} - 4) (x^{2} + 2)^{\frac{7}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{63} (7 x^{2} - 4) (x^{2} + 2)^{\frac{7}{2}} + C

Graph

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