integral of (x^2)/(x^2-5x+6)

解答

\int \frac{x ^{2}}{x ^{2} - 5 x + 6} d x

4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x + C

求解步骤

\int \frac{x ^{2}}{x ^{2} - 5 x + 6} d x

\frac{x ^{2}}{x ^{2} - 5 x + 6} = \frac{5 x - 6}{x ^{2} - 5 x + 6} + 1

= \int \frac{5 x - 6}{x ^{2} - 5 x + 6} + 1 d 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{5 x - 6}{x ^{2} - 5 x + 6} d x + \int 1 d x

\int \frac{5 x - 6}{x ^{2} - 5 x + 6} d x = 4 (5 (\frac{1}{8} ln 4 x^{2} - 20 x + 24 - \frac{5}{8} (ln ∣ 2 x - 4 ∣ - ln ∣ 2 x - 6 ∣)) + 3 (\frac{1}{2} ln ∣ 2 x - 4 ∣ - \frac{1}{2} ln ∣ 2 x - 6 ∣))

\int 1 d x = x

= 4 (5 (\frac{1}{8} ln 4 x^{2} - 20 x + 24 - \frac{5}{8} (ln ∣ 2 x - 4 ∣ - ln ∣ 2 x - 6 ∣)) + 3 (\frac{1}{2} ln ∣ 2 x - 4 ∣ - \frac{1}{2} ln ∣ 2 x - 6 ∣)) + x

乘开

5 (\frac{1}{8} ln 4 x^{2} - 20 x + 24 - \frac{5}{8} (ln ∣ 2 x - 4 ∣ - ln ∣ 2 x - 6 ∣)) + 3 (\frac{1}{2} ln ∣ 2 x - 4 ∣ - \frac{1}{2} ln ∣ 2 x - 6 ∣) : \frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣

= 4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x

解答补常数

= 4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x + C

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