integral of (5x-2)/(x^3+x^2)

解答

\int \frac{5 x - 2}{x ^{3} + x ^{2}} d x

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

求解步骤

\int \frac{5 x - 2}{x ^{3} + x ^{2}} d x

乘开

\frac{5 x - 2}{x ^{3} + x ^{2}} : \frac{5 x}{x ^{3} + x ^{2}} - \frac{2}{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{5 x}{x ^{3} + x ^{2}} d x - \int \frac{2}{x ^{3} + x ^{2}} d x

\int \frac{5 x}{x ^{3} + x ^{2}} d x = 5 (ln ∣ x ∣ - ln ∣ x + 1 ∣)

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

= 5 (ln ∣ x ∣ - ln ∣ x + 1 ∣) - 2 (- ln ∣ x ∣ - \frac{1}{x} + ln ∣ x + 1 ∣)

化简

5 (ln ∣ x ∣ - ln ∣ x + 1 ∣) - 2 (- ln ∣ x ∣ - \frac{1}{x} + ln ∣ x + 1 ∣) : 7 ln ∣ x ∣ - 7 ln ∣ x + 1 ∣ + \frac{2}{x}

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

解答补常数

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

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