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Beliebt Rechnen >

integral of (5x+1)/((x-1)(2x+1))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

\frac{5 x + 1}{( x - 1 ) ( 2 x + 1 )} : \frac{2}{x - 1} + \frac{1}{2 x + 1}

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of (x-4)/(x^2(x+1))

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

integral of 1/(xlog_{5)(x)}

\int \frac{1}{x lo g _{5} ( x )} d x

integral of x^5(ln(x))^2

\int x^{5} (ln (x))^{2} d x

integral of sec^4(x/5)tan^4(x/5)

\int sec^{4} (\frac{x}{5}) tan^{4} (\frac{x}{5}) d x

integral of (50+y)/(5x+1)

\int \frac{50 + y}{5 x + 1}