integral of (x^2-x+36)/(x^3+6x)

Lösung

\int \frac{x ^{2} - x + 36}{x ^{3} + 6 x} d x

6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 6 - \frac{1}{6} arctan (\frac{x}{6}) + C

Schritte zur Lösung

\int \frac{x ^{2} - x + 36}{x ^{3} + 6 x} d x

Ermittle den Partialbruch von

\frac{x ^{2} - x + 36}{x ^{3} + 6 x} : \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 6}

= \int \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 6} 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{6}{x} d x + \int \frac{- 5 x - 1}{x ^{2} + 6} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{- 5 x - 1}{x ^{2} + 6} d x = - \frac{5}{2} ln x^{2} + 6 - \frac{1}{6} arctan (\frac{x}{6})

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 6 - \frac{1}{6} arctan (\frac{x}{6})

Füge eine Konstante zur Lösung hinzu

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 6 - \frac{1}{6} arctan (\frac{x}{6}) + C

\frac{\partial}{\partial x} (10 x (x^{2} + y^{2})^{\frac{1}{2}})

\frac{\partial}{\partial x} (\frac{x y}{1 + x})

\int - cos (d) x d x

n = 0 \sum \infty \frac{1}{2 n + 4}

\int 28 x^{3} - 15 x^{2} + 10 x d x