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Beliebt Rechnen >

integral of (x^2-x+2)/(x^2-3x+5)

Lösung

\int \frac{x ^{2} - x + 2}{x ^{2} - 3 x + 5} d x

Lösung

x + ln 4 x^{2} - 12 x + 20 - \frac{1}{11} arctan (\frac{1}{11} (2 x - 3)) + \frac{1}{11} arctan (\frac{1}{11} (2 x - 3)) + C

Schritte zur Lösung

\int \frac{x ^{2} - x + 2}{x ^{2} - 3 x + 5} d x

Multipliziere aus

\frac{x ^{2} - x + 2}{x ^{2} - 3 x + 5} : \frac{x ^{2}}{x ^{2} - 3 x + 5} - \frac{x}{x ^{2} - 3 x + 5} + \frac{2}{x ^{2} - 3 x + 5}

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 ^{2}}{x ^{2} - 3 x + 5} d x - \int \frac{x}{x ^{2} - 3 x + 5} d x + \int \frac{2}{x ^{2} - 3 x + 5} d x

\int \frac{x ^{2}}{x ^{2} - 3 x + 5} d x = \frac{3}{2} ln 4 x^{2} - 12 x + 20 - \frac{1}{11} arctan (\frac{1}{11} (2 x - 3)) + x

\int \frac{x}{x ^{2} - 3 x + 5} d x = \frac{1}{2} ln 4 x^{2} - 12 x + 20 + \frac{3}{11} arctan (\frac{2 x - 3}{11})

\int \frac{2}{x ^{2} - 3 x + 5} d x = \frac{4}{11} arctan (\frac{2 x - 3}{11})

= \frac{3}{2} ln 4 x^{2} - 12 x + 20 - \frac{1}{11} arctan (\frac{1}{11} (2 x - 3)) + x - (\frac{1}{2} ln 4 x^{2} - 12 x + 20 + \frac{3}{11} arctan (\frac{2 x - 3}{11})) + \frac{4}{11} arctan (\frac{2 x - 3}{11})

Vereinfache

\frac{3}{2} ln 4 x^{2} - 12 x + 20 - \frac{1}{11} arctan (\frac{1}{11} (2 x - 3)) + x - (\frac{1}{2} ln 4 x^{2} - 12 x + 20 + \frac{3}{11} arctan (\frac{2 x - 3}{11})) + \frac{4}{11} arctan (\frac{2 x - 3}{11}) : x + ln 4 x^{2} - 12 x + 20 - \frac{1}{11} arctan (\frac{1}{11} (2 x - 3)) + \frac{1}{11} arctan (\frac{1}{11} (2 x - 3))

= x + ln 4 x^{2} - 12 x + 20 - \frac{1}{11} arctan (\frac{1}{11} (2 x - 3)) + \frac{1}{11} arctan (\frac{1}{11} (2 x - 3))

Füge eine Konstante zur Lösung hinzu

= x + ln 4 x^{2} - 12 x + 20 - \frac{1}{11} arctan (\frac{1}{11} (2 x - 3)) + \frac{1}{11} arctan (\frac{1}{11} (2 x - 3)) + C

Graph

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