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

解答

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

- 3 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ + \frac{2}{x} + C

求解步骤

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

化简

\frac{x ^{2} + 3 x + 2}{x ^{4} - x ^{2}} : \frac{x + 2}{x ^{2} ( x - 1 )}

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

乘开

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

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

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

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

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

化简

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

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

解答补常数

= - 3 ln ∣ x ∣ + 3 ln ∣ x - 1 ∣ + \frac{2}{x} + C

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