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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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