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

解答

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

4 ln ∣ x ∣ + \frac{3}{x} - 4 ln ∣ x + 1 ∣ + C

求解步骤

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

将

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

用部份分式展开:

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

= \int \frac{4}{x} - \frac{3}{x ^{2}} - \frac{4}{x + 1} 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{4}{x} d x - \int \frac{3}{x ^{2}} d x - \int \frac{4}{x + 1} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

\int \frac{4}{x + 1} d x = 4 ln ∣ x + 1 ∣

= 4 ln ∣ x ∣ - (- \frac{3}{x}) - 4 ln ∣ x + 1 ∣

化简

= 4 ln ∣ x ∣ + \frac{3}{x} - 4 ln ∣ x + 1 ∣

解答补常数

= 4 ln ∣ x ∣ + \frac{3}{x} - 4 ln ∣ x + 1 ∣ + C

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