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

解答

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

\frac{3}{2} ln x^{2} + 2 - \frac{1}{2} arctan (\frac{x}{2}) - 2 ln ∣ x ∣ + C

求解步骤

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

乘开

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

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

\int \frac{x}{x ^{3} + 2 x} d x = \frac{1}{2} arctan (\frac{x}{2})

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

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

化简

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

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

解答补常数

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