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Beliebt Rechnen >

integral of (12x^3-3x^2)/(3x^4-x^3)

Lösung

\int \frac{12 x ^{3} - 3 x ^{2}}{3 x ^{4} - x ^{3}} d x

Lösung

ln ∣ 3 x - 1 ∣ + 3 ln (3 ∣ x ∣) + C

Schritte zur Lösung

\int \frac{12 x ^{3} - 3 x ^{2}}{3 x ^{4} - x ^{3}} d x

Multipliziere aus

\frac{12 x ^{3} - 3 x ^{2}}{3 x ^{4} - x ^{3}} : \frac{12 x ^{3}}{3 x ^{4} - x ^{3}} - \frac{3 x ^{2}}{3 x ^{4} - x ^{3}}

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{12 x ^{3}}{3 x ^{4} - x ^{3}} d x - \int \frac{3 x ^{2}}{3 x ^{4} - x ^{3}} d x

\int \frac{12 x ^{3}}{3 x ^{4} - x ^{3}} d x = 4 ln ∣ 3 x - 1 ∣

\int \frac{3 x ^{2}}{3 x ^{4} - x ^{3}} d x = 3 (ln ∣ 3 x - 1 ∣ - ln (3 ∣ x ∣))

= 4 ln ∣ 3 x - 1 ∣ - 3 (ln ∣ 3 x - 1 ∣ - ln (3 ∣ x ∣))

Vereinfache

4 ln ∣ 3 x - 1 ∣ - 3 (ln ∣ 3 x - 1 ∣ - ln (3 ∣ x ∣)) : ln ∣ 3 x - 1 ∣ + 3 ln (3 ∣ x ∣)

= ln ∣ 3 x - 1 ∣ + 3 ln (3 ∣ x ∣)

Füge eine Konstante zur Lösung hinzu

= ln ∣ 3 x - 1 ∣ + 3 ln (3 ∣ x ∣) + C

Graph

Beliebte Beispiele

integral of (2x^4-x-4)/(x^3+8)

\int \frac{2 x ^{4} - x - 4}{x ^{3} + 8} d x

integral of (e^{12t})/(sqrt(4e^{6t)-1)}

\int \frac{e ^{12 t}}{4 e ^{6 t} - 1} d t

integral of (6sqrt(x^2-1))/x

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

integral of 1/(x^2sqrt(2x-4))

\int \frac{1}{x ^{2} 2 x - 4} d x

integral of 2/3 e^{-2t}

\int \frac{2}{3} e^{- 2 t} d t