integral of sqrt(u)(u^2+3u+7)

Lösung

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

\frac{2}{7} u^{\frac{7}{2}} + \frac{6}{5} u^{\frac{5}{2}} + \frac{14}{3} u^{\frac{3}{2}} + C

Schritte zur Lösung

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

Multipliziere aus

u (u^{2} + 3 u + 7) : u^{\frac{5}{2}} + 3 u^{\frac{3}{2}} + 7 u

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

= \int u^{\frac{5}{2}} d u + \int 3 u^{\frac{3}{2}} d u + \int 7 u d u

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

\int 3 u^{\frac{3}{2}} d u = \frac{6}{5} u^{\frac{5}{2}}

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

= \frac{2}{7} u^{\frac{7}{2}} + \frac{6}{5} u^{\frac{5}{2}} + \frac{14}{3} u^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{7} u^{\frac{7}{2}} + \frac{6}{5} u^{\frac{5}{2}} + \frac{14}{3} u^{\frac{3}{2}} + C

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