integral of ((36x^3-109x^2+25)/(3x-5))

Lösung

\int (\frac{36 x ^{3} - 109 x ^{2} + 25}{3 x - 5}) d x

4 x^{3} - \frac{49 x ^{2}}{6} - \frac{245}{9} x - \frac{1000}{27} ln ∣ 3 x - 5 ∣ + C

Schritte zur Lösung

\int \frac{36 x ^{3} - 109 x ^{2} + 25}{3 x - 5} d x

Lange Division

\frac{36 x ^{3} - 109 x ^{2} + 25}{3 x - 5} : 12 x^{2} - \frac{49 x}{3} - \frac{245}{9} - \frac{1000}{9 ( 3 x - 5 )}

= \int 12 x^{2} - \frac{49 x}{3} - \frac{245}{9} - \frac{1000}{9 ( 3 x - 5 )} 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 12 x^{2} d x - \int \frac{49 x}{3} d x - \int \frac{245}{9} d x - \int \frac{1000}{9 ( 3 x - 5 )} d x

\int 12 x^{2} d x = 4 x^{3}

\int \frac{49 x}{3} d x = \frac{49 x ^{2}}{6}

\int \frac{245}{9} d x = \frac{245}{9} x

\int \frac{1000}{9 ( 3 x - 5 )} d x = \frac{1000}{27} ln ∣ 3 x - 5 ∣

= 4 x^{3} - \frac{49 x ^{2}}{6} - \frac{245}{9} x - \frac{1000}{27} ln ∣ 3 x - 5 ∣

Füge eine Konstante zur Lösung hinzu

= 4 x^{3} - \frac{49 x ^{2}}{6} - \frac{245}{9} x - \frac{1000}{27} ln ∣ 3 x - 5 ∣ + C

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