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Beliebt Rechnen >

integral of (x^2+6x+4)/(x^4+8x^2+16)

Lösung

\int \frac{x ^{2} + 6 x + 4}{x ^{4} + 8 x ^{2} + 16} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{2} + 6 x + 4}{x ^{4} + 8 x ^{2} + 16} d x

Ermittle den Partialbruch von

\frac{x ^{2} + 6 x + 4}{x ^{4} + 8 x ^{2} + 16} : \frac{1}{x ^{2} + 4} + \frac{6 x}{( x ^{2} + 4 ) ^{2}}

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

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

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

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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