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Beliebt Rechnen >

integral of (-2x+6)/(x^2+2x+3)

Lösung

\int \frac{- 2 x + 6}{x ^{2} + 2 x + 3} d x

Lösung

- ln x^{2} + 2 x + 3 + 42 arctan (\frac{1}{2} (x + 1)) + C

Schritte zur Lösung

\int \frac{- 2 x + 6}{x ^{2} + 2 x + 3} d x

Multipliziere aus

\frac{- 2 x + 6}{x ^{2} + 2 x + 3} : - \frac{2 x}{x ^{2} + 2 x + 3} + \frac{6}{x ^{2} + 2 x + 3}

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} + 2 x + 3} d x + \int \frac{6}{x ^{2} + 2 x + 3} d x

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

\int \frac{6}{x ^{2} + 2 x + 3} d x = 32 arctan (\frac{1}{2} (x + 1))

= - (ln x^{2} + 2 x + 3 - 2 arctan (\frac{1}{2} (x + 1))) + 32 arctan (\frac{1}{2} (x + 1))

Vereinfache

= - ln x^{2} + 2 x + 3 + 42 arctan (\frac{1}{2} (x + 1))

Füge eine Konstante zur Lösung hinzu

= - ln x^{2} + 2 x + 3 + 42 arctan (\frac{1}{2} (x + 1)) + C

Graph

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