integral of \sqrt[3]{x}(2-x)^2

Lösung

\int 3 x (2 - x)^{2} d x

3 x^{\frac{4}{3}} - \frac{12}{7} x^{\frac{7}{3}} + \frac{3}{10} x^{\frac{10}{3}} + C

Schritte zur Lösung

\int 3 x (2 - x)^{2} d x

Multipliziere aus

3 x (2 - x)^{2} : 4 3 x - 4 x^{\frac{4}{3}} + x^{\frac{7}{3}}

= \int 4 3 x - 4 x^{\frac{4}{3}} + x^{\frac{7}{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 4 3 x d x - \int 4 x^{\frac{4}{3}} d x + \int x^{\frac{7}{3}} d x

\int 4 3 x d x = 3 x^{\frac{4}{3}}

\int 4 x^{\frac{4}{3}} d x = \frac{12}{7} x^{\frac{7}{3}}

\int x^{\frac{7}{3}} d x = \frac{3}{10} x^{\frac{10}{3}}

= 3 x^{\frac{4}{3}} - \frac{12}{7} x^{\frac{7}{3}} + \frac{3}{10} x^{\frac{10}{3}}

Füge eine Konstante zur Lösung hinzu

= 3 x^{\frac{4}{3}} - \frac{12}{7} x^{\frac{7}{3}} + \frac{3}{10} x^{\frac{10}{3}} + C

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