integral of (x^2-22)/(x^3-2x^2+x)

解答

\int \frac{x ^{2} - 22}{x ^{3} - 2 x ^{2} + x} d x

- 22 ln ∣ x ∣ + 23 ln ∣ x - 1 ∣ + \frac{21}{x - 1} + C

求解步骤

\int \frac{x ^{2} - 22}{x ^{3} - 2 x ^{2} + x} d x

将

\frac{x ^{2} - 22}{x ^{3} - 2 x ^{2} + x}

用部份分式展开:

- \frac{22}{x} + \frac{23}{x - 1} - \frac{21}{( x - 1 ) ^{2}}

= \int - \frac{22}{x} + \frac{23}{x - 1} - \frac{21}{( x - 1 ) ^{2}} 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{22}{x} d x + \int \frac{23}{x - 1} d x - \int \frac{21}{( x - 1 ) ^{2}} d x

\int \frac{22}{x} d x = 22 ln ∣ x ∣

\int \frac{23}{x - 1} d x = 23 ln ∣ x - 1 ∣

\int \frac{21}{( x - 1 ) ^{2}} d x = - \frac{21}{x - 1}

= - 22 ln ∣ x ∣ + 23 ln ∣ x - 1 ∣ - (- \frac{21}{x - 1})

化简

= - 22 ln ∣ x ∣ + 23 ln ∣ x - 1 ∣ + \frac{21}{x - 1}

解答补常数

= - 22 ln ∣ x ∣ + 23 ln ∣ x - 1 ∣ + \frac{21}{x - 1} + C

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