integral of (4x+1)/(x^2+2x+3)

Lösung

\int \frac{4 x + 1}{x ^{2} + 2 x + 3} d x

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

Schritte zur Lösung

\int \frac{4 x + 1}{x ^{2} + 2 x + 3} d x

Multipliziere aus

\frac{4 x + 1}{x ^{2} + 2 x + 3} : \frac{4 x}{x ^{2} + 2 x + 3} + \frac{1}{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{4 x}{x ^{2} + 2 x + 3} d x + \int \frac{1}{x ^{2} + 2 x + 3} d x

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

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

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

Füge eine Konstante zur Lösung hinzu

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

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