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Beliebt Rechnen >

integral of 3x*sqrt(x-8)

Lösung

\int 3 x \cdot x - 8 d x

Lösung

\frac{6}{5} (x - 8)^{\frac{5}{2}} + 16 (x - 8)^{\frac{3}{2}} + C

Schritte zur Lösung

\int 3 x x - 8 d x

Entferne die Konstante:

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

= 3 \cdot \int x x - 8 d x

Wende U-Substitution an

= 3 \cdot \int (u + 8) u d u

Multipliziere aus

(u + 8) u : u^{\frac{3}{2}} + 8 u

= 3 \cdot \int u^{\frac{3}{2}} + 8 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

= 3 (\int u^{\frac{3}{2}} d u + \int 8 u d u)

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

\int 8 u d u = \frac{16}{3} u^{\frac{3}{2}}

= 3 (\frac{2}{5} u^{\frac{5}{2}} + \frac{16}{3} u^{\frac{3}{2}})

Setze in

u = x - 8

ein

= 3 (\frac{2}{5} (x - 8)^{\frac{5}{2}} + \frac{16}{3} (x - 8)^{\frac{3}{2}})

Vereinfache

3 (\frac{2}{5} (x - 8)^{\frac{5}{2}} + \frac{16}{3} (x - 8)^{\frac{3}{2}}) : \frac{6}{5} (x - 8)^{\frac{5}{2}} + 16 (x - 8)^{\frac{3}{2}}

= \frac{6}{5} (x - 8)^{\frac{5}{2}} + 16 (x - 8)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{6}{5} (x - 8)^{\frac{5}{2}} + 16 (x - 8)^{\frac{3}{2}} + C

Graph

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