integral of (x+4)(x+3)^{1/4}

Lösung

\int (x + 4) (x + 3)^{\frac{1}{4}} d x

\frac{4}{9} (x + 3)^{\frac{9}{4}} + \frac{4}{5} (x + 3)^{\frac{5}{4}} + C

Schritte zur Lösung

\int (x + 4) (x + 3)^{\frac{1}{4}} d x

Wende U-Substitution an

= \int u (u - 1)^{\frac{1}{4}} d u

Wende U-Substitution an

= \int (v + 1) v^{\frac{1}{4}} d v

Multipliziere aus

(v + 1) v^{\frac{1}{4}} : v^{\frac{5}{4}} + v^{\frac{1}{4}}

= \int v^{\frac{5}{4}} + v^{\frac{1}{4}} 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}{4}} d v + \int v^{\frac{1}{4}} d v

\int v^{\frac{5}{4}} d v = \frac{4}{9} v^{\frac{9}{4}}

\int v^{\frac{1}{4}} d v = \frac{4}{5} v^{\frac{5}{4}}

= \frac{4}{9} v^{\frac{9}{4}} + \frac{4}{5} v^{\frac{5}{4}}

Ersetze zurück

= \frac{4}{9} (x + 4 - 1)^{\frac{9}{4}} + \frac{4}{5} (x + 4 - 1)^{\frac{5}{4}}

Vereinfache

\frac{4}{9} (x + 4 - 1)^{\frac{9}{4}} + \frac{4}{5} (x + 4 - 1)^{\frac{5}{4}} : \frac{4}{9} (x + 3)^{\frac{9}{4}} + \frac{4}{5} (x + 3)^{\frac{5}{4}}

= \frac{4}{9} (x + 3)^{\frac{9}{4}} + \frac{4}{5} (x + 3)^{\frac{5}{4}}

Füge eine Konstante zur Lösung hinzu

= \frac{4}{9} (x + 3)^{\frac{9}{4}} + \frac{4}{5} (x + 3)^{\frac{5}{4}} + C