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Beliebt Rechnen >

integral of (4x^2-7x-12)/(x(x+2)(x-3))

Lösung

\int \frac{4 x ^{2} - 7 x - 12}{x ( x + 2 ) ( x - 3 )} d x

Lösung

2 ln ∣ x ∣ + \frac{9}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ + C

Schritte zur Lösung

\int \frac{4 x ^{2} - 7 x - 12}{x ( x + 2 ) ( x - 3 )} d x

Ermittle den Partialbruch von

\frac{4 x ^{2} - 7 x - 12}{x ( x + 2 ) ( x - 3 )} : \frac{2}{x} + \frac{9}{5 ( x + 2 )} + \frac{1}{5 ( x - 3 )}

= \int \frac{2}{x} + \frac{9}{5 ( x + 2 )} + \frac{1}{5 ( 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{2}{x} d x + \int \frac{9}{5 ( x + 2 )} d x + \int \frac{1}{5 ( x - 3 )} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{9}{5 ( x + 2 )} d x = \frac{9}{5} ln ∣ x + 2 ∣

\int \frac{1}{5 ( x - 3 )} d x = \frac{1}{5} ln ∣ x - 3 ∣

= 2 ln ∣ x ∣ + \frac{9}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x ∣ + \frac{9}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ + C

Graph

Beliebte Beispiele

t a y l or x^{3}, 3

integral of x^{10}e^{x^{11}}

\int x^{10} e^{x^{11}} d x

derivative of 7sin^2(x)

\frac{d}{d x} (7 sin^{2} (x))

derivative of (3x^2-1/(x-2))

\frac{d}{d x} (\frac{3 x ^{2} - 1}{x - 2})

tangent f(x)=sec(x),\at x=(pi/3)

t an g e n t f (x) = sec (x), at x = (\frac{π}{3})