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Beliebt Rechnen >

integral of (2x-4)/(x^2+4x+29)

Lösung

\int \frac{2 x - 4}{x ^{2} + 4 x + 29} d x

Lösung

ln \frac{1}{25} (x + 2)^{2} + 1 - \frac{4}{5} arctan (\frac{1}{5} (x + 2)) - \frac{4}{5} arctan (\frac{1}{5} (x + 2)) + C

Schritte zur Lösung

\int \frac{2 x - 4}{x ^{2} + 4 x + 29} d x

Multipliziere aus

\frac{2 x - 4}{x ^{2} + 4 x + 29} : \frac{2 x}{x ^{2} + 4 x + 29} - \frac{4}{x ^{2} + 4 x + 29}

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 + 29} d x - \int \frac{4}{x ^{2} + 4 x + 29} d x

\int \frac{2 x}{x ^{2} + 4 x + 29} d x = \frac{1}{5} (5 ln \frac{( x + 2 ) ^{2}}{25} + 1 - 4 arctan (\frac{x + 2}{5}))

\int \frac{4}{x ^{2} + 4 x + 29} d x = \frac{4}{5} arctan (\frac{1}{5} (x + 2))

= \frac{1}{5} (5 ln \frac{( x + 2 ) ^{2}}{25} + 1 - 4 arctan (\frac{x + 2}{5})) - \frac{4}{5} arctan (\frac{1}{5} (x + 2))

Vereinfache

\frac{1}{5} (5 ln \frac{( x + 2 ) ^{2}}{25} + 1 - 4 arctan (\frac{x + 2}{5})) - \frac{4}{5} arctan (\frac{1}{5} (x + 2)) : ln \frac{1}{25} (x + 2)^{2} + 1 - \frac{4}{5} arctan (\frac{1}{5} (x + 2)) - \frac{4}{5} arctan (\frac{1}{5} (x + 2))

= ln \frac{1}{25} (x + 2)^{2} + 1 - \frac{4}{5} arctan (\frac{1}{5} (x + 2)) - \frac{4}{5} arctan (\frac{1}{5} (x + 2))

Füge eine Konstante zur Lösung hinzu

= ln \frac{1}{25} (x + 2)^{2} + 1 - \frac{4}{5} arctan (\frac{1}{5} (x + 2)) - \frac{4}{5} arctan (\frac{1}{5} (x + 2)) + C

Graph

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