integral of (x+2)sqrt(x-2)

Lösung

\int (x + 2) x - 2 d x

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

Schritte zur Lösung

\int (x + 2) x - 2 d x

Wende U-Substitution an

= \int u (u + 4) d u

Multipliziere aus

u (u + 4) : u^{\frac{3}{2}} + 4 u

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

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

\int 4 u d u = \frac{8}{3} u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} + \frac{8}{3} u^{\frac{3}{2}}

Setze in

u = x - 2

ein

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

Füge eine Konstante zur Lösung hinzu

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

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