integral of (x-1)^2*(x+1)^2

Lösung

\int (x - 1)^{2} \cdot (x + 1)^{2} d x

- x^{5} + \frac{6 x ^{5}}{5} - \frac{2 x ^{3}}{3} + x + C

Schritte zur Lösung

\int (x - 1)^{2} (x + 1)^{2} d x

Wende die partielle Integration an

= (x - 1)^{2} (x + 1)^{2} x - \int (4 x^{3} - 4 x) x d x

\int (4 x^{3} - 4 x) x d x = 2 x^{3} (x^{2} - 1) - 2 (\frac{3 x ^{5}}{5} - \frac{x ^{3}}{3})

= (x - 1)^{2} (x + 1)^{2} x - (2 x^{3} (x^{2} - 1) - 2 (\frac{3 x ^{5}}{5} - \frac{x ^{3}}{3}))

Vereinfache

(x - 1)^{2} (x + 1)^{2} x - (2 x^{3} (x^{2} - 1) - 2 (\frac{3 x ^{5}}{5} - \frac{x ^{3}}{3})) : - x^{5} + \frac{6 x ^{5}}{5} - \frac{2 x ^{3}}{3} + x

= - x^{5} + \frac{6 x ^{5}}{5} - \frac{2 x ^{3}}{3} + x

Füge eine Konstante zur Lösung hinzu

= - x^{5} + \frac{6 x ^{5}}{5} - \frac{2 x ^{3}}{3} + x + C

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