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Beliebt Rechnen >

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

Lösung

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

Lösung

- \frac{1}{2} ln x^{2} + 1 + ln ∣ x ∣ + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

= - \frac{1}{2} ln x^{2} + 1 + ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{2} ln x^{2} + 1 + ln ∣ x ∣ + C

Graph

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