integral of (x^2-5x+3)/(x-7)

解答

\int \frac{x ^{2} - 5 x + 3}{x - 7} d x

9 x + 17 ln ∣ x - 7 ∣ + \frac{1}{2} (x - 7)^{2} - 63 + C

求解步骤

\int \frac{x ^{2} - 5 x + 3}{x - 7} d x

乘开

\frac{x ^{2} - 5 x + 3}{x - 7} : \frac{x ^{2}}{x - 7} - \frac{5 x}{x - 7} + \frac{3}{x - 7}

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 - 7} d x - \int \frac{5 x}{x - 7} d x + \int \frac{3}{x - 7} d x

\int \frac{x ^{2}}{x - 7} d x = \frac{( x - 7 ) ^{2}}{2} + 14 (x - 7) + 49 ln ∣ x - 7 ∣

\int \frac{5 x}{x - 7} d x = 5 (x - 7 + 7 ln ∣ x - 7 ∣)

\int \frac{3}{x - 7} d x = 3 ln ∣ x - 7 ∣

= \frac{( x - 7 ) ^{2}}{2} + 14 (x - 7) + 49 ln ∣ x - 7 ∣ - 5 (x - 7 + 7 ln ∣ x - 7 ∣) + 3 ln ∣ x - 7 ∣

化简

\frac{( x - 7 ) ^{2}}{2} + 14 (x - 7) + 49 ln ∣ x - 7 ∣ - 5 (x - 7 + 7 ln ∣ x - 7 ∣) + 3 ln ∣ x - 7 ∣ : 9 x + 17 ln ∣ x - 7 ∣ + \frac{1}{2} (x - 7)^{2} - 63

= 9 x + 17 ln ∣ x - 7 ∣ + \frac{1}{2} (x - 7)^{2} - 63

解答补常数

= 9 x + 17 ln ∣ x - 7 ∣ + \frac{1}{2} (x - 7)^{2} - 63 + C

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