integral of 1/(6x^3-7x^2+2x)

Lösung

\int \frac{1}{6 x ^{3} - 7 x ^{2} + 2 x} d x

\frac{1}{2} ln ∣ x ∣ - 2 ln ∣ 2 x - 1 ∣ + \frac{3}{2} ln ∣ 3 x - 2 ∣ + C

Schritte zur Lösung

\int \frac{1}{6 x ^{3} - 7 x ^{2} + 2 x} d x

Ermittle den Partialbruch von

\frac{1}{6 x ^{3} - 7 x ^{2} + 2 x} : \frac{1}{2 x} - \frac{4}{2 x - 1} + \frac{9}{2 ( 3 x - 2 )}

= \int \frac{1}{2 x} - \frac{4}{2 x - 1} + \frac{9}{2 ( 3 x - 2 )} 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 \frac{1}{2 x} d x - \int \frac{4}{2 x - 1} d x + \int \frac{9}{2 ( 3 x - 2 )} d x

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

\int \frac{4}{2 x - 1} d x = 2 ln ∣ 2 x - 1 ∣

\int \frac{9}{2 ( 3 x - 2 )} d x = \frac{3}{2} ln ∣ 3 x - 2 ∣

= \frac{1}{2} ln ∣ x ∣ - 2 ln ∣ 2 x - 1 ∣ + \frac{3}{2} ln ∣ 3 x - 2 ∣

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} ln ∣ x ∣ - 2 ln ∣ 2 x - 1 ∣ + \frac{3}{2} ln ∣ 3 x - 2 ∣ + C

\int \frac{x - 4}{x ( x - 2 ) ( x - 5 )} d x

\int \frac{40}{S ^{2}} d S

\int ln (2 x - 3) d x

\int (x^{3} - x^{2} - x + 1) e^{2 x} d x

\int \frac{1}{4 x ^{2} + 8 x + 19} d x