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Beliebt Rechnen >

integral of (2x^2+x-8)/(x^3-4x)

Lösung

\int \frac{2 x ^{2} + x - 8}{x ^{3} - 4 x} d x

Lösung

2 ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{4} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

\int \frac{2 x ^{2} + x - 8}{x ^{3} - 4 x} d x

Ermittle den Partialbruch von

\frac{2 x ^{2} + x - 8}{x ^{3} - 4 x} : \frac{2}{x} - \frac{1}{4 ( x + 2 )} + \frac{1}{4 ( x - 2 )}

= \int \frac{2}{x} - \frac{1}{4 ( x + 2 )} + \frac{1}{4 ( x - 2 )} 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{1}{4 ( x + 2 )} d x + \int \frac{1}{4 ( x - 2 )} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{1}{4 ( x + 2 )} d x = \frac{1}{4} ln ∣ x + 2 ∣

\int \frac{1}{4 ( x - 2 )} d x = \frac{1}{4} ln ∣ x - 2 ∣

= 2 ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{4} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= 2 ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{4} ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

(2x-1)y^'+y^2=0

(2 x - 1) y^{^{'}} + y^{2} = 0

limit as x approaches 4+of 1

x \to 4 + lim (1)

integral of (sqrt(1+x))/(x^2)

\int \frac{1 + x}{x ^{2}} d x

(\partial)/(\partial x)(3x^4+5y^4+9xy)

\frac{\partial}{\partial x} (3 x^{4} + 5 y^{4} + 9 x y)

y^{''}-5y^'+6y=e^x

y^{^{''}} - 5 y^{^{'}} + 6 y = e^{x}