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Beliebt Rechnen >

integral of sqrt(3+x)*(x+1)^2

Lösung

\int 3 + x \cdot (x + 1)^{2} d x

Lösung

\frac{2}{7} (x + 3)^{\frac{7}{2}} - \frac{8}{5} (x + 3)^{\frac{5}{2}} + \frac{8}{3} (x + 3)^{\frac{3}{2}} + C

Schritte zur Lösung

\int 3 + x (x + 1)^{2} d x

Wende U-Substitution an

= \int u^{2} u + 2 d u

Wende U-Substitution an

= \int (v - 2)^{2} v d v

Multipliziere aus

(v - 2)^{2} v : v^{\frac{5}{2}} - 4 v^{\frac{3}{2}} + 4 v

= \int v^{\frac{5}{2}} - 4 v^{\frac{3}{2}} + 4 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

= \int v^{\frac{5}{2}} d v - \int 4 v^{\frac{3}{2}} d v + \int 4 v d v

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

\int 4 v^{\frac{3}{2}} d v = \frac{8}{5} v^{\frac{5}{2}}

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

= \frac{2}{7} v^{\frac{7}{2}} - \frac{8}{5} v^{\frac{5}{2}} + \frac{8}{3} v^{\frac{3}{2}}

Ersetze zurück

= \frac{2}{7} (x + 1 + 2)^{\frac{7}{2}} - \frac{8}{5} (x + 1 + 2)^{\frac{5}{2}} + \frac{8}{3} (x + 1 + 2)^{\frac{3}{2}}

Vereinfache

\frac{2}{7} (x + 1 + 2)^{\frac{7}{2}} - \frac{8}{5} (x + 1 + 2)^{\frac{5}{2}} + \frac{8}{3} (x + 1 + 2)^{\frac{3}{2}} : \frac{2}{7} (x + 3)^{\frac{7}{2}} - \frac{8}{5} (x + 3)^{\frac{5}{2}} + \frac{8}{3} (x + 3)^{\frac{3}{2}}

= \frac{2}{7} (x + 3)^{\frac{7}{2}} - \frac{8}{5} (x + 3)^{\frac{5}{2}} + \frac{8}{3} (x + 3)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{7} (x + 3)^{\frac{7}{2}} - \frac{8}{5} (x + 3)^{\frac{5}{2}} + \frac{8}{3} (x + 3)^{\frac{3}{2}} + C

Graph

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