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

解答

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

5 x + 3 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 10 + C

求解步骤

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

乘开

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

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

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

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

= \frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣ + x - 2 + 2 ln ∣ x - 2 ∣ - 3 ln ∣ x - 2 ∣

化简

\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣ + x - 2 + 2 ln ∣ x - 2 ∣ - 3 ln ∣ x - 2 ∣ : 5 x + 3 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 10

= 5 x + 3 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 10

解答补常数

= 5 x + 3 ln ∣ x - 2 ∣ + \frac{1}{2} (x - 2)^{2} - 10 + C

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