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Beliebt Rechnen >

integral of (24t+6t^3)/(8t^2+t^4)

Lösung

\int \frac{24 t + 6 t ^{3}}{8 t ^{2} + t ^{4}} d t

Lösung

3 ln (t) + \frac{3}{2} ln t^{2} + 8 + C

Schritte zur Lösung

\int \frac{24 t + 6 t ^{3}}{8 t ^{2} + t ^{4}} d t

Multipliziere aus

\frac{24 t + 6 t ^{3}}{8 t ^{2} + t ^{4}} : \frac{24 t}{8 t ^{2} + t ^{4}} + \frac{6 t ^{3}}{8 t ^{2} + t ^{4}}

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{24 t}{8 t ^{2} + t ^{4}} d t + \int \frac{6 t ^{3}}{8 t ^{2} + t ^{4}} d t

\int \frac{24 t}{8 t ^{2} + t ^{4}} d t = 24 (\frac{1}{8} ln (t) - \frac{1}{16} ln t^{2} + 8)

\int \frac{6 t ^{3}}{8 t ^{2} + t ^{4}} d t = 3 ln t^{2} + 8

= 24 (\frac{1}{8} ln (t) - \frac{1}{16} ln t^{2} + 8) + 3 ln t^{2} + 8

Vereinfache

24 (\frac{1}{8} ln (t) - \frac{1}{16} ln t^{2} + 8) + 3 ln t^{2} + 8 : 3 ln (t) + \frac{3}{2} ln t^{2} + 8

= 3 ln (t) + \frac{3}{2} ln t^{2} + 8

Füge eine Konstante zur Lösung hinzu

= 3 ln (t) + \frac{3}{2} ln t^{2} + 8 + C

Graph

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