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Beliebt Rechnen >

integral of (2x+3)/(x(x-2))

Lösung

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

Lösung

- \frac{3}{2} ln ∣ x ∣ + \frac{7}{2} ln ∣ x - 2 ∣ + C

Schritte zur Lösung

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

Multipliziere aus

\frac{2 x + 3}{x ( x - 2 )} : \frac{2 x}{x ( x - 2 )} + \frac{3}{x ( x - 2 )}

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

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

\int \frac{3}{x ( x - 2 )} d x = - 3 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

= 2 ln ∣ x - 2 ∣ - 3 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

Vereinfache

2 ln ∣ x - 2 ∣ - 3 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) : - \frac{3}{2} ln ∣ x ∣ + \frac{7}{2} ln ∣ x - 2 ∣

= - \frac{3}{2} ln ∣ x ∣ + \frac{7}{2} ln ∣ x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{3}{2} ln ∣ x ∣ + \frac{7}{2} ln ∣ x - 2 ∣ + C

Graph

Beliebte Beispiele

integral of 14xe^{-7x}

\int 14 x e^{- 7 x} d x

integral of 14x^6-12x^3+20x

\int 14 x^{6} - 12 x^{3} + 20 x d x

integral of (3-sqrt(x)+x)

\int (3 - x + x) d x

integral of (sec^2(2x))/(1+3tan^2(2x))

\int \frac{sec ^{2} ( 2 x )}{1 + 3 tan ^{2} ( 2 x )} d x

integral of (x^4)/(\sqrt[5]{x^5+4)}

\int \frac{x ^{4}}{5 x ^{5} + 4} d x