integral of (3x^2+5x+1)/(x-4)

解答

\int \frac{3 x ^{2} + 5 x + 1}{x - 4} d x

29 x + 69 ln ∣ x - 4 ∣ + \frac{3}{2} (x - 4)^{2} - 116 + C

求解步骤

\int \frac{3 x ^{2} + 5 x + 1}{x - 4} d x

乘开

\frac{3 x ^{2} + 5 x + 1}{x - 4} : \frac{3 x ^{2}}{x - 4} + \frac{5 x}{x - 4} + \frac{1}{x - 4}

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{3 x ^{2}}{x - 4} d x + \int \frac{5 x}{x - 4} d x + \int \frac{1}{x - 4} d x

\int \frac{3 x ^{2}}{x - 4} d x = 3 (\frac{( x - 4 ) ^{2}}{2} + 8 (x - 4) + 16 ln ∣ x - 4 ∣)

\int \frac{5 x}{x - 4} d x = 5 (x - 4 + 4 ln ∣ x - 4 ∣)

\int \frac{1}{x - 4} d x = ln ∣ x - 4 ∣

= 3 (\frac{( x - 4 ) ^{2}}{2} + 8 (x - 4) + 16 ln ∣ x - 4 ∣) + 5 (x - 4 + 4 ln ∣ x - 4 ∣) + ln ∣ x - 4 ∣

化简

3 (\frac{( x - 4 ) ^{2}}{2} + 8 (x - 4) + 16 ln ∣ x - 4 ∣) + 5 (x - 4 + 4 ln ∣ x - 4 ∣) + ln ∣ x - 4 ∣ : 29 x + 69 ln ∣ x - 4 ∣ + \frac{3}{2} (x - 4)^{2} - 116

= 29 x + 69 ln ∣ x - 4 ∣ + \frac{3}{2} (x - 4)^{2} - 116

解答补常数

= 29 x + 69 ln ∣ x - 4 ∣ + \frac{3}{2} (x - 4)^{2} - 116 + C

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