integral of (x^2+x)/(x^3-x^2+x-1)

Lösung

\int \frac{x ^{2} + x}{x ^{3} - x ^{2} + x - 1} d x

ln ∣ x - 1 ∣ + arctan (x) + C

Schritte zur Lösung

\int \frac{x ^{2} + x}{x ^{3} - x ^{2} + x - 1} d x

Ermittle den Partialbruch von

\frac{x ^{2} + x}{x ^{3} - x ^{2} + x - 1} : \frac{1}{x - 1} + \frac{1}{x ^{2} + 1}

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

\int \frac{1}{x - 1} d x = ln ∣ x - 1 ∣

\int \frac{1}{x ^{2} + 1} d x = arctan (x)

= ln ∣ x - 1 ∣ + arctan (x)

Füge eine Konstante zur Lösung hinzu

= ln ∣ x - 1 ∣ + arctan (x) + C

y^{^{'}} + \frac{1 - x}{2 x} y = 0

d er i v a t i v e f (x) = 2 x + \frac{1936}{x}

t an g e n t f (x) = 110 x - 16 x^{2}, at x = 4

d er i v a t i v e \frac{1}{x + 8}

\frac{d}{d x} (\frac{e ^{x}}{x ^{2} + 7})