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

解答

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

ln ∣ x ∣ - ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣ + C

求解步骤

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

将

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

用部份分式展开:

\frac{1}{x} - \frac{1}{x + 1} - \frac{1}{x + 2}

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

= ln ∣ x ∣ - ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣

解答补常数

= ln ∣ x ∣ - ln ∣ x + 1 ∣ - ln ∣ x + 2 ∣ + C