integral of (4x-11)/(2x^2-11x+15)

解答

\int \frac{4 x - 11}{2 x ^{2} - 11 x + 15} d x

4 (- \frac{5}{2} ln ∣ 2 x - 5 ∣ + 3 ln ∣ x - 3 ∣ + \frac{5}{2} - 3) + 22 (\frac{1}{2} ln ∣ 4 x - 10 ∣ - \frac{1}{2} ln ∣ 4 x - 12 ∣) + C

求解步骤

\int \frac{4 x - 11}{2 x ^{2} - 11 x + 15} d x

乘开

\frac{4 x - 11}{2 x ^{2} - 11 x + 15} : \frac{4 x}{2 x ^{2} - 11 x + 15} - \frac{11}{2 x ^{2} - 11 x + 15}

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{4 x}{2 x ^{2} - 11 x + 15} d x - \int \frac{11}{2 x ^{2} - 11 x + 15} d x

\int \frac{4 x}{2 x ^{2} - 11 x + 15} d x = 4 (- \frac{5}{2} ln ∣ 2 x - 5 ∣ + 3 ln ∣ x - 3 ∣ + \frac{5}{2} - 3)

\int \frac{11}{2 x ^{2} - 11 x + 15} d x = - 22 (\frac{1}{2} ln ∣ 4 x - 10 ∣ - \frac{1}{2} ln ∣ 4 x - 12 ∣)

= 4 (- \frac{5}{2} ln ∣ 2 x - 5 ∣ + 3 ln ∣ x - 3 ∣ + \frac{5}{2} - 3) - (- 22 (\frac{1}{2} ln ∣ 4 x - 10 ∣ - \frac{1}{2} ln ∣ 4 x - 12 ∣))

化简

= 4 (- \frac{5}{2} ln ∣ 2 x - 5 ∣ + 3 ln ∣ x - 3 ∣ + \frac{5}{2} - 3) + 22 (\frac{1}{2} ln ∣ 4 x - 10 ∣ - \frac{1}{2} ln ∣ 4 x - 12 ∣)

解答补常数

= 4 (- \frac{5}{2} ln ∣ 2 x - 5 ∣ + 3 ln ∣ x - 3 ∣ + \frac{5}{2} - 3) + 22 (\frac{1}{2} ln ∣ 4 x - 10 ∣ - \frac{1}{2} ln ∣ 4 x - 12 ∣) + C

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