integral of (9x^3-8x+6)/(x^3-x^2)

解答

\int \frac{9 x ^{3} - 8 x + 6}{x ^{3} - x ^{2}} d x

\frac{9 x ^{2}}{x - 1} + 7 ln ∣ x - 1 ∣ - \frac{9}{x - 1} + 2 ln ∣ x ∣ + \frac{6}{x} + C

求解步骤

\int \frac{9 x ^{3} - 8 x + 6}{x ^{3} - x ^{2}} d x

乘开

\frac{9 x ^{3} - 8 x + 6}{x ^{3} - x ^{2}} : \frac{9 x ^{3}}{x ^{3} - x ^{2}} - \frac{8 x}{x ^{3} - x ^{2}} + \frac{6}{x ^{3} - x ^{2}}

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{9 x ^{3}}{x ^{3} - x ^{2}} d x - \int \frac{8 x}{x ^{3} - x ^{2}} d x + \int \frac{6}{x ^{3} - x ^{2}} d x

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

\int \frac{8 x}{x ^{3} - x ^{2}} d x = 8 (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

\int \frac{6}{x ^{3} - x ^{2}} d x = 6 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣)

= 9 (\frac{x ^{2}}{x - 1} + ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - 8 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) + 6 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣)

化简

9 (\frac{x ^{2}}{x - 1} + ln ∣ x - 1 ∣ - \frac{1}{x - 1}) - 8 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) + 6 (- ln ∣ x ∣ + \frac{1}{x} + ln ∣ x - 1 ∣) : \frac{9 x ^{2}}{x - 1} + 7 ln ∣ x - 1 ∣ - \frac{9}{x - 1} + 2 ln ∣ x ∣ + \frac{6}{x}

= \frac{9 x ^{2}}{x - 1} + 7 ln ∣ x - 1 ∣ - \frac{9}{x - 1} + 2 ln ∣ x ∣ + \frac{6}{x}

解答补常数

= \frac{9 x ^{2}}{x - 1} + 7 ln ∣ x - 1 ∣ - \frac{9}{x - 1} + 2 ln ∣ x ∣ + \frac{6}{x} + C

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