integral of (x^2-11x-18)/(x(x^2+3x+3))

Lösung

\int \frac{x ^{2} - 11 x - 18}{x ( x ^{2} + 3 x + 3 )} d x

- 6 ln ∣ x ∣ + 14 (\frac{1}{4} ln 4 x^{2} + 12 x + 12 - \frac{3}{2} arctan (\frac{2}{3} x + 3)) + \frac{14}{3} arctan (\frac{2}{3} x + 3) + C

Schritte zur Lösung

\int \frac{x ^{2} - 11 x - 18}{x ( x ^{2} + 3 x + 3 )} d x

Multipliziere aus

x (x^{2} + 3 x + 3) : x^{3} + 3 x^{2} + 3 x

= \int \frac{x ^{2} - 11 x - 18}{x ^{3} + 3 x ^{2} + 3 x} d x

Ermittle den Partialbruch von

\frac{x ^{2} - 11 x - 18}{x ^{3} + 3 x ^{2} + 3 x} : - \frac{6}{x} + \frac{7 x + 7}{x ^{2} + 3 x + 3}

= \int - \frac{6}{x} + \frac{7 x + 7}{x ^{2} + 3 x + 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{6}{x} d x + \int \frac{7 x + 7}{x ^{2} + 3 x + 3} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{7 x + 7}{x ^{2} + 3 x + 3} d x = 14 (\frac{1}{4} ln 4 x^{2} + 12 x + 12 - \frac{3}{2} arctan (\frac{2}{3} x + 3)) + \frac{14}{3} arctan (\frac{2}{3} x + 3)

= - 6 ln ∣ x ∣ + 14 (\frac{1}{4} ln 4 x^{2} + 12 x + 12 - \frac{3}{2} arctan (\frac{2}{3} x + 3)) + \frac{14}{3} arctan (\frac{2}{3} x + 3)

Füge eine Konstante zur Lösung hinzu

= - 6 ln ∣ x ∣ + 14 (\frac{1}{4} ln 4 x^{2} + 12 x + 12 - \frac{3}{2} arctan (\frac{2}{3} x + 3)) + \frac{14}{3} arctan (\frac{2}{3} x + 3) + C

\frac{\partial}{\partial x} (5 x y - 4 x^{2} y)

\frac{\partial}{\partial x} (x)

x \to 0 lim (sin^{4} (x))

l a pl a ce t r an s f or m cos (2 (t - \frac{5 π}{2}))

s l o p e (2.3) (1.3)