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Beliebt Rechnen >

integral of (8+2w+4w^2)/((w^2+2)^2w)

Lösung

\int \frac{8 + 2 w + 4 w ^{2}}{( w ^{2} + 2 ) ^{2} w} d w

Lösung

- ln w^{2} + 2 + \frac{1}{2 2} (arctan (\frac{w}{2}) + \frac{1}{2} sin (2 arctan (\frac{w}{2}))) + 2 ln ∣ w ∣ + C

Schritte zur Lösung

\int \frac{8 + 2 w + 4 w ^{2}}{( w ^{2} + 2 ) ^{2} w} d w

Ermittle den Partialbruch von

\frac{8 + 2 w + 4 w ^{2}}{( w ^{2} + 2 ) ^{2} w} : - \frac{2 w}{w ^{2} + 2} + \frac{2}{( w ^{2} + 2 ) ^{2}} + \frac{2}{w}

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

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{2 w}{w ^{2} + 2} d w + \int \frac{2}{( w ^{2} + 2 ) ^{2}} d w + \int \frac{2}{w} d w

\int \frac{2 w}{w ^{2} + 2} d w = ln w^{2} + 2

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

\int \frac{2}{w} d w = 2 ln ∣ w ∣

= - ln w^{2} + 2 + \frac{1}{2 2} (arctan (\frac{w}{2}) + \frac{1}{2} sin (2 arctan (\frac{w}{2}))) + 2 ln ∣ w ∣

Füge eine Konstante zur Lösung hinzu

= - ln w^{2} + 2 + \frac{1}{2 2} (arctan (\frac{w}{2}) + \frac{1}{2} sin (2 arctan (\frac{w}{2}))) + 2 ln ∣ w ∣ + C

Graph

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