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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Ermittle den Partialbruch von

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

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

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

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

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

= 2 (- \frac{1}{x} - arctan (x))

Füge eine Konstante zur Lösung hinzu

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

Graph

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