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

Lösung

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

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{2}{x ^{2}} d x = - \frac{2}{x}

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

= - 2 ln ∣ x ∣ - (- \frac{2}{x}) + 2 ln ∣ x - 2 ∣

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

(ln (1 - x))^{^{'}}

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

\int \frac{1}{n ln ( n )}

l a pl a ce t r an s f or m t^{3} e^{t}

\int x^{2} + 4 x d x