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Beliebt Rechnen >

integral of (s^2-1)/((s^2+1)^2)

Lösung

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

Lösung

- \frac{s}{1 + s ^{2}} + C

Schritte zur Lösung

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

Ermittle den Partialbruch von

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

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

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

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

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

= arctan (s) - (arctan (s) + \frac{s}{1 + s ^{2}})

Vereinfache

= - \frac{s}{1 + s ^{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{s}{1 + s ^{2}} + C

Graph

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