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Beliebt Rechnen >

integral of (16s+16)/((s^2+1)(s-1)^3)

Lösung

\int \frac{16 s + 16}{( s ^{2} + 1 ) ( s - 1 ) ^{3}} d s

Lösung

8 arctan (s) + \frac{8}{s - 1} - \frac{8}{( s - 1 ) ^{2}} + C

Schritte zur Lösung

\int \frac{16 s + 16}{( s ^{2} + 1 ) ( s - 1 ) ^{3}} d s

Ermittle den Partialbruch von

\frac{16 s + 16}{( s ^{2} + 1 ) ( s - 1 ) ^{3}} : \frac{8}{s ^{2} + 1} - \frac{8}{( s - 1 ) ^{2}} + \frac{16}{( s - 1 ) ^{3}}

= \int \frac{8}{s ^{2} + 1} - \frac{8}{( s - 1 ) ^{2}} + \frac{16}{( s - 1 ) ^{3}} 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{8}{s ^{2} + 1} d s - \int \frac{8}{( s - 1 ) ^{2}} d s + \int \frac{16}{( s - 1 ) ^{3}} d s

\int \frac{8}{s ^{2} + 1} d s = 8 arctan (s)

\int \frac{8}{( s - 1 ) ^{2}} d s = - \frac{8}{s - 1}

\int \frac{16}{( s - 1 ) ^{3}} d s = - \frac{8}{( s - 1 ) ^{2}}

= 8 arctan (s) - (- \frac{8}{s - 1}) - \frac{8}{( s - 1 ) ^{2}}

Vereinfache

= 8 arctan (s) + \frac{8}{s - 1} - \frac{8}{( s - 1 ) ^{2}}

Füge eine Konstante zur Lösung hinzu

= 8 arctan (s) + \frac{8}{s - 1} - \frac{8}{( s - 1 ) ^{2}} + C

Graph

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