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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

\frac{2 x - 1}{x ^{2} - 1} : \frac{2 x}{x ^{2} - 1} - \frac{1}{x ^{2} - 1}

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of sec^2(x)tan^1(x)

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

integral of (5.4145)/(sqrt(1+0.91x))

\int \frac{5.4145}{1 + 0.91 x} d x

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

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

integral of x/(a^2-x^2)

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

integral of (ln((2a)/x))^2

\int (ln (\frac{2 a}{x}))^{2} d x