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

解答

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

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

求解步骤

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

乘开

\frac{x ^{3} - x + 1}{x ^{2} ( x - 1 ) ^{3}} : \frac{x}{( x - 1 ) ^{3}} - \frac{1}{x ( x - 1 ) ^{3}} + \frac{1}{x ^{2} ( x - 1 ) ^{3}}

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

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

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

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

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

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

化简

- \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}} - (- ln ∣ x ∣ + ln ∣ x - 1 ∣ + \frac{1}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}}) - 3 ln ∣ x ∣ + \frac{1}{x} + 3 ln ∣ x - 1 ∣ + \frac{2}{x - 1} - \frac{1}{2 ( x - 1 ) ^{2}} : - \frac{1}{2 ( x - 1 ) ^{2}} - 2 ln ∣ x ∣ + 2 ln ∣ x - 1 ∣ + \frac{1}{x}

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

解答补常数

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

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