integral of (2x^2-x+20)/((x-2)(x^2+9))

解答

\int \frac{2 x ^{2} - x + 20}{( x - 2 ) ( x ^{2} + 9 )} d x

2 ln ∣ x - 2 ∣ + \frac{1}{13} ln x^{2} + 9 - \frac{49}{39} arctan (\frac{x}{3}) - \frac{10}{13} ln \frac{x ^{2}}{9} + 1 + \frac{1}{13} (9 ln x^{2} + 9 + 12 arctan (\frac{x}{3})) + C

求解步骤

\int \frac{2 x ^{2} - x + 20}{( x - 2 ) ( x ^{2} + 9 )} d x

乘开

\frac{2 x ^{2} - x + 20}{( x - 2 ) ( x ^{2} + 9 )} : \frac{2 x ^{2}}{( x - 2 ) ( x ^{2} + 9 )} - \frac{x}{( x - 2 ) ( x ^{2} + 9 )} + \frac{20}{( x - 2 ) ( x ^{2} + 9 )}

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 ^{2}}{( x - 2 ) ( x ^{2} + 9 )} d x - \int \frac{x}{( x - 2 ) ( x ^{2} + 9 )} d x + \int \frac{20}{( x - 2 ) ( x ^{2} + 9 )} d x

\int \frac{2 x ^{2}}{( x - 2 ) ( x ^{2} + 9 )} d x = 2 (\frac{4}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (\frac{9}{2} ln x^{2} + 9 + 6 arctan (\frac{x}{3})))

\int \frac{x}{( x - 2 ) ( x ^{2} + 9 )} d x = \frac{2}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3}))

\int \frac{20}{( x - 2 ) ( x ^{2} + 9 )} d x = \frac{20}{13} ln ∣ x - 2 ∣ - \frac{10}{13} ln \frac{x ^{2}}{9} + 1 - \frac{40}{39} arctan (\frac{x}{3})

= 2 (\frac{4}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (\frac{9}{2} ln x^{2} + 9 + 6 arctan (\frac{x}{3}))) - (\frac{2}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3}))) + \frac{20}{13} ln ∣ x - 2 ∣ - \frac{10}{13} ln \frac{x ^{2}}{9} + 1 - \frac{40}{39} arctan (\frac{x}{3})

化简

2 (\frac{4}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (\frac{9}{2} ln x^{2} + 9 + 6 arctan (\frac{x}{3}))) - (\frac{2}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3}))) + \frac{20}{13} ln ∣ x - 2 ∣ - \frac{10}{13} ln \frac{x ^{2}}{9} + 1 - \frac{40}{39} arctan (\frac{x}{3}) : 2 ln ∣ x - 2 ∣ + \frac{1}{13} ln x^{2} + 9 - \frac{49}{39} arctan (\frac{x}{3}) - \frac{10}{13} ln \frac{x ^{2}}{9} + 1 + \frac{1}{13} (9 ln x^{2} + 9 + 12 arctan (\frac{x}{3}))

= 2 ln ∣ x - 2 ∣ + \frac{1}{13} ln x^{2} + 9 - \frac{49}{39} arctan (\frac{x}{3}) - \frac{10}{13} ln \frac{x ^{2}}{9} + 1 + \frac{1}{13} (9 ln x^{2} + 9 + 12 arctan (\frac{x}{3}))

解答补常数

= 2 ln ∣ x - 2 ∣ + \frac{1}{13} ln x^{2} + 9 - \frac{49}{39} arctan (\frac{x}{3}) - \frac{10}{13} ln \frac{x ^{2}}{9} + 1 + \frac{1}{13} (9 ln x^{2} + 9 + 12 arctan (\frac{x}{3})) + C

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