integral of 6xsqrt(x-2)

Lösung

\int 6 x x - 2 d x

6 (\frac{2}{5} (x - 2)^{\frac{5}{2}} + \frac{4}{3} (x - 2)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int 6 x x - 2 d x

Entferne die Konstante:

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

= 6 \cdot \int x x - 2 d x

Wende U-Substitution an

= 6 \cdot \int (u + 2) u d u

Multipliziere aus

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

= 6 \cdot \int u^{\frac{3}{2}} + 2 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

= 6 (\int u^{\frac{3}{2}} d u + \int 2 u d u)

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

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

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

Setze in

u = x - 2

ein

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

Füge eine Konstante zur Lösung hinzu

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

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