integral of (3x^2+12x-20)/(x^4-8x^2+16)

解答

\int \frac{3 x ^{2} + 12 x - 20}{x ^{4} - 8 x ^{2} + 16} d x

- ln ∣ x + 2 ∣ + ln ∣ x - 2 ∣ + \frac{3}{2} arctan (\frac{x ^{2}}{2 2}) + \frac{1}{2 ( x + 2 )} + \frac{1}{2 ( x - 2 )} + C

求解步骤

\int \frac{3 x ^{2} + 12 x - 20}{x ^{4} - 8 x ^{2} + 16} d x

乘开

\frac{3 x ^{2} + 12 x - 20}{x ^{4} - 8 x ^{2} + 16} : \frac{3 x ^{2}}{x ^{4} - 8 x ^{2} + 16} + \frac{12 x}{x ^{4} - 8 x ^{2} + 16} - \frac{20}{x ^{4} - 8 x ^{2} + 16}

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} - 8 x ^{2} + 16} d x + \int \frac{12 x}{x ^{4} - 8 x ^{2} + 16} d x - \int \frac{20}{x ^{4} - 8 x ^{2} + 16} d x

\int \frac{3 x ^{2}}{x ^{4} - 8 x ^{2} + 16} d x = 3 (- \frac{1}{8} ln ∣ x + 2 ∣ - \frac{1}{4 ( x + 2 )} + \frac{1}{8} ln ∣ x - 2 ∣ - \frac{1}{4 ( x - 2 )})

\int \frac{12 x}{x ^{4} - 8 x ^{2} + 16} d x = \frac{3}{2} arctan (\frac{x ^{2}}{2 2})

\int \frac{20}{x ^{4} - 8 x ^{2} + 16} d x = 20 (\frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{16 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{16 ( x - 2 )})

= 3 (- \frac{1}{8} ln ∣ x + 2 ∣ - \frac{1}{4 ( x + 2 )} + \frac{1}{8} ln ∣ x - 2 ∣ - \frac{1}{4 ( x - 2 )}) + \frac{3}{2} arctan (\frac{x ^{2}}{2 2}) - 20 (\frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{16 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{16 ( x - 2 )})

化简

3 (- \frac{1}{8} ln ∣ x + 2 ∣ - \frac{1}{4 ( x + 2 )} + \frac{1}{8} ln ∣ x - 2 ∣ - \frac{1}{4 ( x - 2 )}) + \frac{3}{2} arctan (\frac{x ^{2}}{2 2}) - 20 (\frac{1}{32} ln ∣ x + 2 ∣ - \frac{1}{16 ( x + 2 )} - \frac{1}{32} ln ∣ x - 2 ∣ - \frac{1}{16 ( x - 2 )}) : - ln ∣ x + 2 ∣ + ln ∣ x - 2 ∣ + \frac{3}{2} arctan (\frac{x ^{2}}{2 2}) + \frac{1}{2 ( x + 2 )} + \frac{1}{2 ( x - 2 )}

= - ln ∣ x + 2 ∣ + ln ∣ x - 2 ∣ + \frac{3}{2} arctan (\frac{x ^{2}}{2 2}) + \frac{1}{2 ( x + 2 )} + \frac{1}{2 ( x - 2 )}

解答补常数

= - ln ∣ x + 2 ∣ + ln ∣ x - 2 ∣ + \frac{3}{2} arctan (\frac{x ^{2}}{2 2}) + \frac{1}{2 ( x + 2 )} + \frac{1}{2 ( x - 2 )} + C

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