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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

\frac{4 x + 1}{x ^{2} + 2 x + 2} : \frac{4 x}{x ^{2} + 2 x + 2} + \frac{1}{x ^{2} + 2 x + 2}

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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