integral of (3x-2)/(5x^2-3x+1)

Lösung

\int \frac{3 x - 2}{5 x ^{2} - 3 x + 1} d x

\frac{6}{11} x arctan (\frac{1}{11} (10 x - 3)) - \frac{4}{11} arctan (\frac{1}{11} (10 x - 3)) - \frac{6}{11} x arctan (\frac{1}{11} (10 x - 3)) + \frac{9}{5 11} arctan (\frac{1}{11} (10 x - 3)) + \frac{3}{10} ln 100 x^{2} - 60 x + 20 + C

Schritte zur Lösung

\int \frac{3 x - 2}{5 x ^{2} - 3 x + 1} d x

Wende die partielle Integration an

= \frac{2}{11} arctan (\frac{1}{11} (10 x - 3)) (3 x - 2) - \int \frac{6}{11} arctan (\frac{1}{11} (10 x - 3)) d x

\int \frac{6}{11} arctan (\frac{1}{11} (10 x - 3)) d x = \frac{3}{10 11} (2 arctan (\frac{10 x - 3}{11}) (10 x - 3) - 11 ln 100 x^{2} - 60 x + 20)

= \frac{2}{11} arctan (\frac{1}{11} (10 x - 3)) (3 x - 2) - \frac{3}{10 11} (2 arctan (\frac{10 x - 3}{11}) (10 x - 3) - 11 ln 100 x^{2} - 60 x + 20)

Vereinfache

\frac{2}{11} arctan (\frac{1}{11} (10 x - 3)) (3 x - 2) - \frac{3}{10 11} (2 arctan (\frac{10 x - 3}{11}) (10 x - 3) - 11 ln 100 x^{2} - 60 x + 20) : \frac{6}{11} x arctan (\frac{1}{11} (10 x - 3)) - \frac{4}{11} arctan (\frac{1}{11} (10 x - 3)) - \frac{6}{11} x arctan (\frac{1}{11} (10 x - 3)) + \frac{9}{5 11} arctan (\frac{1}{11} (10 x - 3)) + \frac{3}{10} ln 100 x^{2} - 60 x + 20

= \frac{6}{11} x arctan (\frac{1}{11} (10 x - 3)) - \frac{4}{11} arctan (\frac{1}{11} (10 x - 3)) - \frac{6}{11} x arctan (\frac{1}{11} (10 x - 3)) + \frac{9}{5 11} arctan (\frac{1}{11} (10 x - 3)) + \frac{3}{10} ln 100 x^{2} - 60 x + 20

Füge eine Konstante zur Lösung hinzu

= \frac{6}{11} x arctan (\frac{1}{11} (10 x - 3)) - \frac{4}{11} arctan (\frac{1}{11} (10 x - 3)) - \frac{6}{11} x arctan (\frac{1}{11} (10 x - 3)) + \frac{9}{5 11} arctan (\frac{1}{11} (10 x - 3)) + \frac{3}{10} ln 100 x^{2} - 60 x + 20 + C

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