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Beliebt Rechnen >

integral of (x^2+x+2)/(x^4+5x^2+4)

Lösung

\int \frac{x ^{2} + x + 2}{x ^{4} + 5 x ^{2} + 4} d x

Lösung

\frac{1}{3} arctan (x) + \frac{1}{3} arctan (\frac{x}{2}) + \frac{1}{6} ln x^{2} + 1 - \frac{1}{6} ln x^{2} + 4 + C

Schritte zur Lösung

\int \frac{x ^{2} + x + 2}{x ^{4} + 5 x ^{2} + 4} d x

Faktorisiere

\frac{x ^{2} + x + 2}{x ^{4} + 5 x ^{2} + 4}

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

Multipliziere aus

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

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

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

\int \frac{x}{( x ^{2} + 1 ) ( x ^{2} + 4 )} d x = \frac{1}{6} ln x^{2} + 1 - \frac{1}{6} ln x^{2} + 4

\int \frac{2}{( x ^{2} + 1 ) ( x ^{2} + 4 )} d x = 2 (\frac{1}{3} arctan (x) - \frac{1}{6} arctan (\frac{x}{2}))

= - \frac{1}{3} arctan (x) + \frac{2}{3} arctan (\frac{x}{2}) + \frac{1}{6} ln x^{2} + 1 - \frac{1}{6} ln x^{2} + 4 + 2 (\frac{1}{3} arctan (x) - \frac{1}{6} arctan (\frac{x}{2}))

Vereinfache

- \frac{1}{3} arctan (x) + \frac{2}{3} arctan (\frac{x}{2}) + \frac{1}{6} ln x^{2} + 1 - \frac{1}{6} ln x^{2} + 4 + 2 (\frac{1}{3} arctan (x) - \frac{1}{6} arctan (\frac{x}{2})) : \frac{1}{3} arctan (x) + \frac{1}{3} arctan (\frac{x}{2}) + \frac{1}{6} ln x^{2} + 1 - \frac{1}{6} ln x^{2} + 4

= \frac{1}{3} arctan (x) + \frac{1}{3} arctan (\frac{x}{2}) + \frac{1}{6} ln x^{2} + 1 - \frac{1}{6} ln x^{2} + 4

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} arctan (x) + \frac{1}{3} arctan (\frac{x}{2}) + \frac{1}{6} ln x^{2} + 1 - \frac{1}{6} ln x^{2} + 4 + C

Graph

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