integral (x^2-x)^4

Lösung

integral

(x^{2} - x)^{4}

\frac{x ^{9}}{9} - \frac{x ^{8}}{2} + \frac{6 x ^{7}}{7} - \frac{2 x ^{6}}{3} + \frac{x ^{5}}{5} + C

Schritte zur Lösung

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

Faktorisiere

x^{2} - x : - (- x^{2} + x)

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

Vereinfache

(- (- x^{2} + x))^{4} : (- x^{2} + x)^{4}

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

Multipliziere aus

(- x^{2} + x)^{4} : x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} + x^{4}

= \int x^{8} - 4 x^{7} + 6 x^{6} - 4 x^{5} + x^{4} d x

Wende die Summenregel an:

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

= \int x^{8} d x - \int 4 x^{7} d x + \int 6 x^{6} d x - \int 4 x^{5} d x + \int x^{4} d x

\int x^{8} d x = \frac{x ^{9}}{9}

\int 4 x^{7} d x = \frac{x ^{8}}{2}

\int 6 x^{6} d x = \frac{6 x ^{7}}{7}

\int 4 x^{5} d x = \frac{2 x ^{6}}{3}

\int x^{4} d x = \frac{x ^{5}}{5}

= \frac{x ^{9}}{9} - \frac{x ^{8}}{2} + \frac{6 x ^{7}}{7} - \frac{2 x ^{6}}{3} + \frac{x ^{5}}{5}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{9}}{9} - \frac{x ^{8}}{2} + \frac{6 x ^{7}}{7} - \frac{2 x ^{6}}{3} + \frac{x ^{5}}{5} + C

in t e g r a l \frac{2}{( 2 x + 3 )}

in t e g r a l \frac{e ^{x}}{e ^{x} + 1}

in t e g r a l \frac{e ^{- x}}{( 1 + e ^{- x} )}

in t e g r a l x sin^{2} (x)

in t e g r a l \frac{x ^{2}}{9 - x ^{2}}