integral of+(3x^2-21x+32)/(x^3-8x^2+16x)

解答

\int + \frac{3 x ^{2} - 21 x + 32}{x ^{3} - 8 x ^{2} + 16 x} d x

ln ∣ x - 4 ∣ + \frac{1}{x - 4} + 2 ln ∣ x ∣ + C

求解步骤

\int \frac{3 x ^{2} - 21 x + 32}{x ^{3} - 8 x ^{2} + 16 x} d x

乘开

\frac{3 x ^{2} - 21 x + 32}{x ^{3} - 8 x ^{2} + 16 x} : \frac{3 x ^{2}}{x ^{3} - 8 x ^{2} + 16 x} - \frac{21 x}{x ^{3} - 8 x ^{2} + 16 x} + \frac{32}{x ^{3} - 8 x ^{2} + 16 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{3 x ^{2}}{x ^{3} - 8 x ^{2} + 16 x} d x - \int \frac{21 x}{x ^{3} - 8 x ^{2} + 16 x} d x + \int \frac{32}{x ^{3} - 8 x ^{2} + 16 x} d x

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

\int \frac{21 x}{x ^{3} - 8 x ^{2} + 16 x} d x = - \frac{21}{x - 4}

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

= 3 (ln ∣ x - 4 ∣ - \frac{4}{x - 4}) - (- \frac{21}{x - 4}) + 32 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{16} ln ∣ x - 4 ∣ - \frac{1}{4 ( x - 4 )})

化简

3 (ln ∣ x - 4 ∣ - \frac{4}{x - 4}) - (- \frac{21}{x - 4}) + 32 (\frac{1}{16} ln ∣ x ∣ - \frac{1}{16} ln ∣ x - 4 ∣ - \frac{1}{4 ( x - 4 )}) : ln ∣ x - 4 ∣ + \frac{1}{x - 4} + 2 ln ∣ x ∣

= ln ∣ x - 4 ∣ + \frac{1}{x - 4} + 2 ln ∣ x ∣

解答补常数

= ln ∣ x - 4 ∣ + \frac{1}{x - 4} + 2 ln ∣ x ∣ + C

in t e g r a l \frac{x}{x ^{2} + 2}

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