integral of 2xsqrt(x-7)

Lösung

\int 2 x x - 7 d x

2 (\frac{2}{5} (x - 7)^{\frac{5}{2}} + \frac{14}{3} (x - 7)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int 2 x x - 7 d x

Entferne die Konstante:

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

= 2 \cdot \int x x - 7 d x

Wende U-Substitution an

= 2 \cdot \int (u + 7) u d u

Multipliziere aus

(u + 7) u : u^{\frac{3}{2}} + 7 u

= 2 \cdot \int u^{\frac{3}{2}} + 7 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

= 2 (\int u^{\frac{3}{2}} d u + \int 7 u d u)

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

\int 7 u d u = \frac{14}{3} u^{\frac{3}{2}}

= 2 (\frac{2}{5} u^{\frac{5}{2}} + \frac{14}{3} u^{\frac{3}{2}})

Setze in

u = x - 7

ein

= 2 (\frac{2}{5} (x - 7)^{\frac{5}{2}} + \frac{14}{3} (x - 7)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{2}{5} (x - 7)^{\frac{5}{2}} + \frac{14}{3} (x - 7)^{\frac{3}{2}}) + C

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