integral of \sqrt[3]{t}(t+1)^3

Lösung

\int 3 t (t + 1)^{3} d t

\frac{3}{13} t^{\frac{13}{3}} + \frac{9}{10} t^{\frac{10}{3}} + \frac{9}{7} t^{\frac{7}{3}} + \frac{3}{4} t^{\frac{4}{3}} + C

Schritte zur Lösung

\int 3 t (t + 1)^{3} d t

Multipliziere aus

3 t (t + 1)^{3} : t^{\frac{10}{3}} + 3 t^{\frac{7}{3}} + 3 t^{\frac{4}{3}} + 3 t

= \int t^{\frac{10}{3}} + 3 t^{\frac{7}{3}} + 3 t^{\frac{4}{3}} + 3 t d t

Wende die Summenregel an:

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

= \int t^{\frac{10}{3}} d t + \int 3 t^{\frac{7}{3}} d t + \int 3 t^{\frac{4}{3}} d t + \int 3 t d t

\int t^{\frac{10}{3}} d t = \frac{3}{13} t^{\frac{13}{3}}

\int 3 t^{\frac{7}{3}} d t = \frac{9}{10} t^{\frac{10}{3}}

\int 3 t^{\frac{4}{3}} d t = \frac{9}{7} t^{\frac{7}{3}}

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

= \frac{3}{13} t^{\frac{13}{3}} + \frac{9}{10} t^{\frac{10}{3}} + \frac{9}{7} t^{\frac{7}{3}} + \frac{3}{4} t^{\frac{4}{3}}

Füge eine Konstante zur Lösung hinzu

= \frac{3}{13} t^{\frac{13}{3}} + \frac{9}{10} t^{\frac{10}{3}} + \frac{9}{7} t^{\frac{7}{3}} + \frac{3}{4} t^{\frac{4}{3}} + C

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