integral of x(1-x)^5

Lösung

\int x (1 - x)^{5} d x

\frac{1}{2} x^{2} (1 - x)^{5} + \frac{1}{42} (35 x^{3} - 105 x^{4} + 126 x^{5} - 70 x^{6} + 15 x^{7}) + C

Schritte zur Lösung

\int x (1 - x)^{5} d x

Wende die partielle Integration an

= \frac{1}{2} x^{2} (1 - x)^{5} - \int - \frac{5}{2} x^{2} (1 - x)^{4} d x

\int - \frac{5}{2} x^{2} (1 - x)^{4} d x = - \frac{1}{42} (35 x^{3} - 105 x^{4} + 126 x^{5} - 70 x^{6} + 15 x^{7})

= \frac{1}{2} x^{2} (1 - x)^{5} - (- \frac{1}{42} (35 x^{3} - 105 x^{4} + 126 x^{5} - 70 x^{6} + 15 x^{7}))

Vereinfache

= \frac{1}{2} x^{2} (1 - x)^{5} + \frac{1}{42} (35 x^{3} - 105 x^{4} + 126 x^{5} - 70 x^{6} + 15 x^{7})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} (1 - x)^{5} + \frac{1}{42} (35 x^{3} - 105 x^{4} + 126 x^{5} - 70 x^{6} + 15 x^{7}) + C

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