integral of (2x+1)/(x^2-6x+13)

Lösung

\int \frac{2 x + 1}{x ^{2} - 6 x + 13} d x

ln x^{2} - 6 x + 13 + 3 arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} arctan (\frac{x - 3}{2}) + C

Schritte zur Lösung

\int \frac{2 x + 1}{x ^{2} - 6 x + 13} d x

Vereinfache

x^{2} : xx

= \int \frac{2 x + 1}{xx - 6 x + 13} d x

Multipliziere aus

\frac{2 x + 1}{xx - 6 x + 13} : \frac{2 x}{xx - 6 x + 13} + \frac{1}{xx - 6 x + 13}

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{2 x}{xx - 6 x + 13} d x + \int \frac{1}{xx - 6 x + 13} d x

\int \frac{2 x}{xx - 6 x + 13} d x = ln x^{2} - 6 x + 13 + 3 arctan (\frac{1}{2} (x - 3))

\int \frac{1}{xx - 6 x + 13} d x = \frac{1}{2} arctan (\frac{x - 3}{2})

= ln x^{2} - 6 x + 13 + 3 arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} arctan (\frac{x - 3}{2})

Füge eine Konstante zur Lösung hinzu

= ln x^{2} - 6 x + 13 + 3 arctan (\frac{1}{2} (x - 3)) + \frac{1}{2} arctan (\frac{x - 3}{2}) + C

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