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

解答

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

2 ln ∣ x - 3 ∣ + ln ∣ x + 1 ∣ - 2 ln ∣ x - 1 ∣ + C

求解步骤

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

将

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

用部份分式展开:

\frac{2}{x - 3} + \frac{1}{x + 1} - \frac{2}{x - 1}

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

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

\int \frac{1}{x + 1} d x = ln ∣ x + 1 ∣

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

= 2 ln ∣ x - 3 ∣ + ln ∣ x + 1 ∣ - 2 ln ∣ x - 1 ∣

解答补常数

= 2 ln ∣ x - 3 ∣ + ln ∣ x + 1 ∣ - 2 ln ∣ x - 1 ∣ + C

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

\int 5 + 12 x - 9 x^{2} d x

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

\int 60 x d x

\int \frac{e ^{2 x}}{7 e ^{2 x} - 11} d x