integral of (x^2+12x-9)/(x^3-9x)

Lösung

\int \frac{x ^{2} + 12 x - 9}{x ^{3} - 9 x} d x

ln ∣ x ∣ - 2 ln ∣ x + 3 ∣ + 2 ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{x ^{2} + 12 x - 9}{x ^{3} - 9 x} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 12 x - 9}{x ^{3} - 9 x} : \frac{1}{x} - \frac{2}{x + 3} + \frac{2}{x - 3}

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{2}{x + 3} d x = 2 ln ∣ x + 3 ∣

\int \frac{2}{x - 3} d x = 2 ln ∣ x - 3 ∣

= ln ∣ x ∣ - 2 ln ∣ x + 3 ∣ + 2 ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= ln ∣ x ∣ - 2 ln ∣ x + 3 ∣ + 2 ln ∣ x - 3 ∣ + C

y^{^{''}} + 13 y^{^{'}} + 36 y = 0, y (0) = 1, y^{^{'}} (0) = 0

\int - 6 sin^{2} (t) d t

\frac{d}{d x} (x^{2} (x - 1))

\int 1 - t^{- 0.52} d t

t an g e n t x - 3