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Beliebt Rechnen >

integral of (3x-4)/(x^2-1)

Lösung

\int \frac{3 x - 4}{x ^{2} - 1} d x

Lösung

\frac{3}{2} ln x^{2} - 1 + 4 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣) + C

Schritte zur Lösung

\int \frac{3 x - 4}{x ^{2} - 1} d x

Multipliziere aus

\frac{3 x - 4}{x ^{2} - 1} : \frac{3 x}{x ^{2} - 1} - \frac{4}{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{3 x}{x ^{2} - 1} d x - \int \frac{4}{x ^{2} - 1} d x

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

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

= \frac{3}{2} ln x^{2} - 1 - (- 4 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣))

Vereinfache

= \frac{3}{2} ln x^{2} - 1 + 4 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣)

Füge eine Konstante zur Lösung hinzu

= \frac{3}{2} ln x^{2} - 1 + 4 (\frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣) + C

Graph

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\int lo g_{7} (x) d x

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