integral of (5x^2-11x+5)/(x^3-4x^2+5x-2)

解答

\int \frac{5 x ^{2} - 11 x + 5}{x ^{3} - 4 x ^{2} + 5 x - 2} d x

2 ln ∣ x - 1 ∣ - \frac{1}{x - 1} + 3 ln ∣ x - 2 ∣ + C

求解步骤

\int \frac{5 x ^{2} - 11 x + 5}{x ^{3} - 4 x ^{2} + 5 x - 2} d x

将

\frac{5 x ^{2} - 11 x + 5}{x ^{3} - 4 x ^{2} + 5 x - 2}

用部份分式展开:

\frac{2}{x - 1} + \frac{1}{( x - 1 ) ^{2}} + \frac{3}{x - 2}

= \int \frac{2}{x - 1} + \frac{1}{( x - 1 ) ^{2}} + \frac{3}{x - 2} 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{2}{x - 1} d x + \int \frac{1}{( x - 1 ) ^{2}} d x + \int \frac{3}{x - 2} d x

\int \frac{2}{x - 1} d x = 2 ln ∣ x - 1 ∣

\int \frac{1}{( x - 1 ) ^{2}} d x = - \frac{1}{x - 1}

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

= 2 ln ∣ x - 1 ∣ - \frac{1}{x - 1} + 3 ln ∣ x - 2 ∣

解答补常数

= 2 ln ∣ x - 1 ∣ - \frac{1}{x - 1} + 3 ln ∣ x - 2 ∣ + C

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

x \to 2 lim (\frac{1}{x ^{3}})

\int_{- 1}^{3} (- 3 x^{2} + 4 x - 5) d x

\int 6 (x + 2) ln (x + 2) d x

t an g e n t y = \frac{1}{4 + 3 x}, (2, \frac{1}{10})