integral of (x^4-5x^3+7x)/x

Lösung

\int \frac{x ^{4} - 5 x ^{3} + 7 x}{x} d x

\frac{x ^{4}}{4} - \frac{5 x ^{3}}{3} + 7 x + C

Schritte zur Lösung

\int \frac{x ^{4} - 5 x ^{3} + 7 x}{x} d x

Vereinfache

\frac{x ^{4} - 5 x ^{3} + 7 x}{x} : x^{3} - 5 x^{2} + 7

= \int x^{3} - 5 x^{2} + 7 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^{3} d x - \int 5 x^{2} d x + \int 7 d x

\int x^{3} d x = \frac{x ^{4}}{4}

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

\int 7 d x = 7 x

= \frac{x ^{4}}{4} - \frac{5 x ^{3}}{3} + 7 x

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{4}}{4} - \frac{5 x ^{3}}{3} + 7 x + C

x \to 6 lim (\frac{g ( x )}{f ( x )})

\int \frac{cos ^{2} ( x )}{sin ^{2} ( x )} d x

\frac{d}{d x} (7 x^{2} - 5 x + 3)

d er i v a t i v e \frac{( 3 e ^{3 x} )}{3}

\int \frac{1}{x ^{2} - 5 x + 4} d x