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

Lösung

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

\frac{x ^{5}}{5} + \frac{3}{5} x^{\frac{10}{3}} + \frac{3}{5} x^{\frac{5}{3}} + C

Schritte zur Lösung

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

Multipliziere aus

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

= \int x^{4} + 2 x^{\frac{7}{3}} + 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 x^{4} d x + \int 2 x^{\frac{7}{3}} d x + \int x^{\frac{2}{3}} d x

\int x^{4} d x = \frac{x ^{5}}{5}

\int 2 x^{\frac{7}{3}} d x = \frac{3}{5} x^{\frac{10}{3}}

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

= \frac{x ^{5}}{5} + \frac{3}{5} x^{\frac{10}{3}} + \frac{3}{5} x^{\frac{5}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{5}}{5} + \frac{3}{5} x^{\frac{10}{3}} + \frac{3}{5} x^{\frac{5}{3}} + C

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