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Beliebt Rechnen >

integral of (x-3)/(x^2-1)

Lösung

\int \frac{x - 3}{x ^{2} - 1} d x

Lösung

\frac{1}{2} ln x^{2} - 1 + 3 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣) + C

Schritte zur Lösung

\int \frac{x - 3}{x ^{2} - 1} d x

Multipliziere aus

\frac{x - 3}{x ^{2} - 1} : \frac{x}{x ^{2} - 1} - \frac{3}{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{x}{x ^{2} - 1} d x - \int \frac{3}{x ^{2} - 1} d x

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative sqrt(x^3)

d er i v a t i v e x^{3}

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

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

limit as x approaches 1 of (x^4-1)/(x^2)

x \to 1 lim (\frac{x ^{4} - 1}{x ^{2}})

derivative f(x)=(2x^2-4)^{-1}

d er i v a t i v e f (x) = (2 x^{2} - 4)^{- 1}

limit as x approaches 8-of x/(x^2-64)

x \to 8 - lim (\frac{x}{x ^{2} - 64})