integral of (x+3)(x+1)^{1/2}

Lösung

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

\frac{2}{3} x (x + 1)^{\frac{3}{2}} - \frac{4}{15} (x + 1)^{\frac{5}{2}} + 2 (x + 1)^{\frac{3}{2}} + C

Schritte zur Lösung

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

Multipliziere aus

(x + 3) (x + 1)^{\frac{1}{2}} : x (x + 1)^{\frac{1}{2}} + 3 (x + 1)^{\frac{1}{2}}

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int x (x + 1)^{\frac{1}{2}} d x = \frac{2}{3} x (x + 1)^{\frac{3}{2}} - \frac{4}{15} (x + 1)^{\frac{5}{2}}

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

= \frac{2}{3} x (x + 1)^{\frac{3}{2}} - \frac{4}{15} (x + 1)^{\frac{5}{2}} + 2 (x + 1)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{3} x (x + 1)^{\frac{3}{2}} - \frac{4}{15} (x + 1)^{\frac{5}{2}} + 2 (x + 1)^{\frac{3}{2}} + C

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