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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

= \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

= \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)

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

integral of 3/(x^3+1)

\int \frac{3}{x ^{3} + 1} d x

integral of (x^3)/(9+x^2)

\int \frac{x ^{3}}{9 + x ^{2}} d x

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\int sin (5 x) sin (7 x) d x

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\int x sin (\frac{π}{2} x) d x

integral of 1/(x(x^6+4))

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