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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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