integral of x(x-7)^7

Lösung

\int x (x - 7)^{7} d x

\frac{1}{2} x^{2} (x - 7)^{7} - \frac{7}{72} (4 x^{9} - 189 x^{8} + 3780 x^{7} - 41160 x^{6} + 259308 x^{5} - 907578 x^{4} + 1411788 x^{3}) + C

Schritte zur Lösung

\int x (x - 7)^{7} d x

Wende die partielle Integration an

= \frac{1}{2} x^{2} (x - 7)^{7} - \int \frac{7}{2} x^{2} (x - 7)^{6} d x

\int \frac{7}{2} x^{2} (x - 7)^{6} d x = \frac{7}{72} (4 x^{9} - 189 x^{8} + 3780 x^{7} - 41160 x^{6} + 259308 x^{5} - 907578 x^{4} + 1411788 x^{3})

= \frac{1}{2} x^{2} (x - 7)^{7} - \frac{7}{72} (4 x^{9} - 189 x^{8} + 3780 x^{7} - 41160 x^{6} + 259308 x^{5} - 907578 x^{4} + 1411788 x^{3})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} (x - 7)^{7} - \frac{7}{72} (4 x^{9} - 189 x^{8} + 3780 x^{7} - 41160 x^{6} + 259308 x^{5} - 907578 x^{4} + 1411788 x^{3}) + C

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