integral of x(x^2-4)^8

Lösung

\int x (x^{2} - 4)^{8} d x

\frac{1}{2} (x^{2} (x^{2} - 4)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 4)^{7} - \frac{1}{18} (7 x^{18} - 189 x^{16} + 2160 x^{14} - 13440 x^{12} + 48384 x^{10} - 96768 x^{8} + 86016 x^{6}))) + C

Schritte zur Lösung

\int x (x^{2} - 4)^{8} d x

Wende U-Substitution an

= \int \frac{( u - 4 ) ^{8}}{2} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int (u - 4)^{8} d u

Wende die partielle Integration an

= \frac{1}{2} ((u - 4)^{8} u - \int 8 (u - 4)^{7} u d u)

\int 8 (u - 4)^{7} u d u = 8 (\frac{1}{2} u^{2} (u - 4)^{7} - \frac{1}{18} (7 u^{9} - 189 u^{8} + 2160 u^{7} - 13440 u^{6} + 48384 u^{5} - 96768 u^{4} + 86016 u^{3}))

= \frac{1}{2} ((u - 4)^{8} u - 8 (\frac{1}{2} u^{2} (u - 4)^{7} - \frac{1}{18} (7 u^{9} - 189 u^{8} + 2160 u^{7} - 13440 u^{6} + 48384 u^{5} - 96768 u^{4} + 86016 u^{3})))

Setze in

u = x^{2}

ein

= \frac{1}{2} ((x^{2} - 4)^{8} x^{2} - 8 (\frac{1}{2} (x^{2})^{2} (x^{2} - 4)^{7} - \frac{1}{18} (7 (x^{2})^{9} - 189 (x^{2})^{8} + 2160 (x^{2})^{7} - 13440 (x^{2})^{6} + 48384 (x^{2})^{5} - 96768 (x^{2})^{4} + 86016 (x^{2})^{3})))

Vereinfache

\frac{1}{2} ((x^{2} - 4)^{8} x^{2} - 8 (\frac{1}{2} (x^{2})^{2} (x^{2} - 4)^{7} - \frac{1}{18} (7 (x^{2})^{9} - 189 (x^{2})^{8} + 2160 (x^{2})^{7} - 13440 (x^{2})^{6} + 48384 (x^{2})^{5} - 96768 (x^{2})^{4} + 86016 (x^{2})^{3}))) : \frac{1}{2} (x^{2} (x^{2} - 4)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 4)^{7} - \frac{1}{18} (7 x^{18} - 189 x^{16} + 2160 x^{14} - 13440 x^{12} + 48384 x^{10} - 96768 x^{8} + 86016 x^{6})))

= \frac{1}{2} (x^{2} (x^{2} - 4)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 4)^{7} - \frac{1}{18} (7 x^{18} - 189 x^{16} + 2160 x^{14} - 13440 x^{12} + 48384 x^{10} - 96768 x^{8} + 86016 x^{6})))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (x^{2} (x^{2} - 4)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 4)^{7} - \frac{1}{18} (7 x^{18} - 189 x^{16} + 2160 x^{14} - 13440 x^{12} + 48384 x^{10} - 96768 x^{8} + 86016 x^{6}))) + C

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