integral of xsqrt(x+15)

Lösung

\int x x + 15 d x

\frac{2}{5} (x + 15)^{\frac{5}{2}} - 10 (x + 15)^{\frac{3}{2}} + C

Schritte zur Lösung

\int x x + 15 d x

Wende U-Substitution an

= \int (u - 15) u d u

Multipliziere aus

(u - 15) u : u^{\frac{3}{2}} - 15 u

= \int u^{\frac{3}{2}} - 15 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 15 u d u

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

\int 15 u d u = 10 u^{\frac{3}{2}}

= \frac{2}{5} u^{\frac{5}{2}} - 10 u^{\frac{3}{2}}

Setze in

u = x + 15

ein

= \frac{2}{5} (x + 15)^{\frac{5}{2}} - 10 (x + 15)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{5} (x + 15)^{\frac{5}{2}} - 10 (x + 15)^{\frac{3}{2}} + C

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