integral of (x^2-x+6)/(x^3+3x)

Lösung

\int \frac{x ^{2} - x + 6}{x ^{3} + 3 x} d x

2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3}) + C

Schritte zur Lösung

\int \frac{x ^{2} - x + 6}{x ^{3} + 3 x} d x

Ermittle den Partialbruch von

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{- x - 1}{x ^{2} + 3} d x = - \frac{1}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3})

= 2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3})

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 3 - \frac{1}{3} arctan (\frac{x}{3}) + C

\frac{\partial}{\partial x} (\frac{x \cdot y}{x + y})

\frac{\partial}{\partial x} (x^{5} + x y^{7} + 4)

\frac{d}{d x} ((ln (x))^{l n (x)})

x \to 7 lim (\frac{x + 4}{x - 7})

d er i v a t i v e f (2) = \frac{x ^{2} - 4}{x - 2}