integral of (x^4+3x+1)/(x^3+x^2-2x)

解答

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

\frac{x ^{2}}{2} - x + \frac{2}{3} ln ∣ x - 1 ∣ + \frac{17}{6} ln ∣ x + 2 ∣ - \frac{1}{2} ln ∣ x ∣ - ln \frac{2}{3} x + \frac{4}{3} + ln \frac{2}{3} x - \frac{2}{3} + C

求解步骤

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

乘开

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

\int \frac{x ^{4}}{x ^{3} + x ^{2} - 2 x} d x = \frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣

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

\int \frac{1}{x ^{3} + x ^{2} - 2 x} d x = - \frac{1}{2} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x + 2 ∣

= \frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣ - ln \frac{4}{3} + \frac{2}{3} x + ln - \frac{2}{3} + \frac{2}{3} x - \frac{1}{2} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x + 2 ∣

化简

\frac{x ^{2}}{2} - x + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{8}{3} ln ∣ x + 2 ∣ - ln \frac{4}{3} + \frac{2}{3} x + ln - \frac{2}{3} + \frac{2}{3} x - \frac{1}{2} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 1 ∣ + \frac{1}{6} ln ∣ x + 2 ∣ : \frac{x ^{2}}{2} - x + \frac{2}{3} ln ∣ x - 1 ∣ + \frac{17}{6} ln ∣ x + 2 ∣ - \frac{1}{2} ln ∣ x ∣ - ln \frac{2}{3} x + \frac{4}{3} + ln \frac{2}{3} x - \frac{2}{3}

= \frac{x ^{2}}{2} - x + \frac{2}{3} ln ∣ x - 1 ∣ + \frac{17}{6} ln ∣ x + 2 ∣ - \frac{1}{2} ln ∣ x ∣ - ln \frac{2}{3} x + \frac{4}{3} + ln \frac{2}{3} x - \frac{2}{3}

解答补常数

= \frac{x ^{2}}{2} - x + \frac{2}{3} ln ∣ x - 1 ∣ + \frac{17}{6} ln ∣ x + 2 ∣ - \frac{1}{2} ln ∣ x ∣ - ln \frac{2}{3} x + \frac{4}{3} + ln \frac{2}{3} x - \frac{2}{3} + C

d er i v a t i v e f (x) = - 5 x^{3} - 7 x^{2} + 8 x - 7

\int x (x + 2)^{2} d x

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\frac{\partial}{\partial x} (x^{2} ln (x^{2} + y^{2}))

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