integral of sqrt(10+\sqrt{x)}

Lösung

\int 10 + x d x

2 (\frac{2}{5} (10 + x)^{\frac{5}{2}} - \frac{20}{3} (10 + x)^{\frac{3}{2}}) + C

Schritte zur Lösung

\int 10 + x d x

Wende U-Substitution an

= \int 10 + u \cdot 2 u d u

Entferne die Konstante:

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

= 2 \cdot \int 10 + u u d u

Wende U-Substitution an

= 2 \cdot \int (v - 10) v d v

Multipliziere aus

(v - 10) v : v^{\frac{3}{2}} - 10 v

= 2 \cdot \int v^{\frac{3}{2}} - 10 v d v

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 v^{\frac{3}{2}} d v - \int 10 v d v)

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

\int 10 v d v = \frac{20}{3} v^{\frac{3}{2}}

= 2 (\frac{2}{5} v^{\frac{5}{2}} - \frac{20}{3} v^{\frac{3}{2}})

Ersetze zurück

= 2 (\frac{2}{5} (10 + x)^{\frac{5}{2}} - \frac{20}{3} (10 + x)^{\frac{3}{2}})

Füge eine Konstante zur Lösung hinzu

= 2 (\frac{2}{5} (10 + x)^{\frac{5}{2}} - \frac{20}{3} (10 + x)^{\frac{3}{2}}) + C