integral of (12x^3-24x^2+5)/(x^2-2x)

Lösung

\int \frac{12 x ^{3} - 24 x ^{2} + 5}{x ^{2} - 2 x} d x

6 x^{2} - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

Schritte zur Lösung

\int \frac{12 x ^{3} - 24 x ^{2} + 5}{x ^{2} - 2 x} d x

Lange Division

\frac{12 x ^{3} - 24 x ^{2} + 5}{x ^{2} - 2 x} : 12 x + \frac{5}{x ^{2} - 2 x}

= \int 12 x + \frac{5}{x ^{2} - 2 x} 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 d x + \int \frac{5}{x ^{2} - 2 x} d x

\int 12 x d x = 6 x^{2}

\int \frac{5}{x ^{2} - 2 x} d x = - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

= 6 x^{2} - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

Füge eine Konstante zur Lösung hinzu

= 6 x^{2} - 5 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) + C

\frac{\partial}{\partial x} (x + 4 y)

\int e^{2 x} 2 x d x

\frac{\partial}{\partial x} (x^{2} - y)

d er i v a t i v e f (x) = x^{3} arcsin (x^{2} e^{x})

in v erse l a pl a ce \frac{s - 2}{s ^{2} + 5 s + 6}