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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

\frac{3 x - 2}{x ^{2} - 4} : \frac{3 x}{x ^{2} - 4} - \frac{2}{x ^{2} - 4}

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

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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