integral of (x+5)/(x^2-6x-15)

Lösung

\int \frac{x + 5}{x ^{2} - 6 x - 15} d x

\frac{1}{2} ln x^{2} - 6 x - 15 - \frac{3}{4 2} (ln \frac{1}{2 6} (x - 3) + 1 - ln \frac{1}{2 6} (x - 3) - 1) - \frac{5}{4 6} (ln \frac{x - 3}{2 6} + 1 - ln \frac{x - 3}{2 6} - 1) + C

Schritte zur Lösung

\int \frac{x + 5}{x ^{2} - 6 x - 15} d x

Multipliziere aus

\frac{x + 5}{x ^{2} - 6 x - 15} : \frac{x}{x ^{2} - 6 x - 15} + \frac{5}{x ^{2} - 6 x - 15}

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} - 6 x - 15} d x + \int \frac{5}{x ^{2} - 6 x - 15} d x

\int \frac{x}{x ^{2} - 6 x - 15} d x = \frac{1}{2} ln x^{2} - 6 x - 15 - \frac{3}{4 2} (ln \frac{1}{2 6} (x - 3) + 1 - ln \frac{1}{2 6} (x - 3) - 1)

\int \frac{5}{x ^{2} - 6 x - 15} d x = - \frac{5}{4 6} (ln \frac{x - 3}{2 6} + 1 - ln \frac{x - 3}{2 6} - 1)

= \frac{1}{2} ln x^{2} - 6 x - 15 - \frac{3}{4 2} (ln \frac{1}{2 6} (x - 3) + 1 - ln \frac{1}{2 6} (x - 3) - 1) - \frac{5}{4 6} (ln \frac{x - 3}{2 6} + 1 - ln \frac{x - 3}{2 6} - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln x^{2} - 6 x - 15 - \frac{3}{4 2} (ln \frac{1}{2 6} (x - 3) + 1 - ln \frac{1}{2 6} (x - 3) - 1) - \frac{5}{4 6} (ln \frac{x - 3}{2 6} + 1 - ln \frac{x - 3}{2 6} - 1) + C

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