integral of (1+x)/(\sqrt[3]{x)}

Lösung

\int \frac{1 + x}{3 x} d x

\frac{3}{2} x^{\frac{2}{3}} + \frac{3}{5} x^{\frac{5}{3}} + C

Schritte zur Lösung

\int \frac{1 + x}{3 x} d x

Multipliziere aus

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

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

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

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

= \frac{3}{2} x^{\frac{2}{3}} + \frac{3}{5} x^{\frac{5}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{2} x^{\frac{2}{3}} + \frac{3}{5} x^{\frac{5}{3}} + C

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