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Beliebt Rechnen >

integral of (-9x-9)/(x(x^2-9))

Lösung

\int \frac{- 9 x - 9}{x ( x ^{2} - 9 )} d x

Lösung

\frac{3}{2} ln \frac{x}{3} + 1 - \frac{3}{2} ln \frac{x}{3} - 1 + ln (x) - \frac{1}{2} ln x^{2} - 9 + C

Schritte zur Lösung

\int \frac{- 9 x - 9}{x ( x ^{2} - 9 )} d x

Multipliziere aus

\frac{- 9 x - 9}{x ( x ^{2} - 9 )} : - \frac{9 x}{x ( x ^{2} - 9 )} - \frac{9}{x ( x ^{2} - 9 )}

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{9 x}{x ( x ^{2} - 9 )} d x - \int \frac{9}{x ( x ^{2} - 9 )} d x

\int \frac{9 x}{x ( x ^{2} - 9 )} d x = - 3 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1)

\int \frac{9}{x ( x ^{2} - 9 )} d x = \frac{1}{2} (- 2 ln (x) + ln x^{2} - 9)

= - (- 3 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1)) - \frac{1}{2} (- 2 ln (x) + ln x^{2} - 9)

Vereinfache

- (- 3 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1)) - \frac{1}{2} (- 2 ln (x) + ln x^{2} - 9) : \frac{3}{2} ln \frac{x}{3} + 1 - \frac{3}{2} ln \frac{x}{3} - 1 + ln (x) - \frac{1}{2} ln x^{2} - 9

= \frac{3}{2} ln \frac{x}{3} + 1 - \frac{3}{2} ln \frac{x}{3} - 1 + ln (x) - \frac{1}{2} ln x^{2} - 9

Füge eine Konstante zur Lösung hinzu

= \frac{3}{2} ln \frac{x}{3} + 1 - \frac{3}{2} ln \frac{x}{3} - 1 + ln (x) - \frac{1}{2} ln x^{2} - 9 + C

Graph

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