integral of (2x+5)/(x^2-4x+13)

Lösung

\int \frac{2 x + 5}{x ^{2} - 4 x + 13} d x

ln \frac{1}{9} (x - 2)^{2} + 1 + \frac{4}{3} arctan (\frac{1}{3} (x - 2)) + \frac{5}{3} arctan (\frac{1}{3} (x - 2)) + C

Schritte zur Lösung

\int \frac{2 x + 5}{x ^{2} - 4 x + 13} d x

Multipliziere aus

\frac{2 x + 5}{x ^{2} - 4 x + 13} : \frac{2 x}{x ^{2} - 4 x + 13} + \frac{5}{x ^{2} - 4 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}{x ^{2} - 4 x + 13} d x + \int \frac{5}{x ^{2} - 4 x + 13} d x

\int \frac{2 x}{x ^{2} - 4 x + 13} d x = \frac{1}{3} (3 ln \frac{( x - 2 ) ^{2}}{9} + 1 + 4 arctan (\frac{x - 2}{3}))

\int \frac{5}{x ^{2} - 4 x + 13} d x = \frac{5}{3} arctan (\frac{1}{3} (x - 2))

= \frac{1}{3} (3 ln \frac{( x - 2 ) ^{2}}{9} + 1 + 4 arctan (\frac{x - 2}{3})) + \frac{5}{3} arctan (\frac{1}{3} (x - 2))

Vereinfache

\frac{1}{3} (3 ln \frac{( x - 2 ) ^{2}}{9} + 1 + 4 arctan (\frac{x - 2}{3})) + \frac{5}{3} arctan (\frac{1}{3} (x - 2)) : ln \frac{1}{9} (x - 2)^{2} + 1 + \frac{4}{3} arctan (\frac{1}{3} (x - 2)) + \frac{5}{3} arctan (\frac{1}{3} (x - 2))

= ln \frac{1}{9} (x - 2)^{2} + 1 + \frac{4}{3} arctan (\frac{1}{3} (x - 2)) + \frac{5}{3} arctan (\frac{1}{3} (x - 2))

Füge eine Konstante zur Lösung hinzu

= ln \frac{1}{9} (x - 2)^{2} + 1 + \frac{4}{3} arctan (\frac{1}{3} (x - 2)) + \frac{5}{3} arctan (\frac{1}{3} (x - 2)) + C

\int \frac{4}{x x ^{2} - 1} d x

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