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

解答

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

ln ∣ 3 x - 1 ∣ + 3 ln (3 ∣ x ∣) + C

求解步骤

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

乘开

\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}}

使用积分加法定则:

\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 ∣))

化简

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 ∣)

解答补常数

= ln ∣ 3 x - 1 ∣ + 3 ln (3 ∣ x ∣) + C

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