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Beliebt Rechnen >

integral of x^2sqrt(8+x)

Lösung

\int x^{2} 8 + x d x

Lösung

\frac{2}{7} (8 + x)^{\frac{7}{2}} - \frac{32}{5} (8 + x)^{\frac{5}{2}} + \frac{128}{3} (8 + x)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x^{2} 8 + x d x

Wende U-Substitution an

= \int (u - 8)^{2} u d u

Multipliziere aus

(u - 8)^{2} u : u^{\frac{5}{2}} - 16 u^{\frac{3}{2}} + 64 u

= \int u^{\frac{5}{2}} - 16 u^{\frac{3}{2}} + 64 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

= \int u^{\frac{5}{2}} d u - \int 16 u^{\frac{3}{2}} d u + \int 64 u d u

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

\int 16 u^{\frac{3}{2}} d u = \frac{32}{5} u^{\frac{5}{2}}

\int 64 u d u = \frac{128}{3} u^{\frac{3}{2}}

= \frac{2}{7} u^{\frac{7}{2}} - \frac{32}{5} u^{\frac{5}{2}} + \frac{128}{3} u^{\frac{3}{2}}

Setze in

u = 8 + x

ein

= \frac{2}{7} (8 + x)^{\frac{7}{2}} - \frac{32}{5} (8 + x)^{\frac{5}{2}} + \frac{128}{3} (8 + x)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{7} (8 + x)^{\frac{7}{2}} - \frac{32}{5} (8 + x)^{\frac{5}{2}} + \frac{128}{3} (8 + x)^{\frac{3}{2}} + C

Graph

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