integral of (9x^2-3x+19)/((x^2+4)(x-3))

解答

\int \frac{9 x ^{2} - 3 x + 19}{( x ^{2} + 4 ) ( x - 3 )} d x

ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2}) + 7 ln ∣ x - 3 ∣ + C

求解步骤

\int \frac{9 x ^{2} - 3 x + 19}{( x ^{2} + 4 ) ( x - 3 )} d x

乘开

\frac{9 x ^{2} - 3 x + 19}{( x ^{2} + 4 ) ( x - 3 )} : \frac{9 x ^{2}}{( x ^{2} + 4 ) ( x - 3 )} - \frac{3 x}{( x ^{2} + 4 ) ( x - 3 )} + \frac{19}{( x ^{2} + 4 ) ( x - 3 )}

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{9 x ^{2}}{( x ^{2} + 4 ) ( x - 3 )} d x - \int \frac{3 x}{( x ^{2} + 4 ) ( x - 3 )} d x + \int \frac{19}{( x ^{2} + 4 ) ( x - 3 )} d x

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

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

\int \frac{19}{( x ^{2} + 4 ) ( x - 3 )} d x = 19 (\frac{1}{13} (- \frac{1}{2} ln x^{2} + 4 - \frac{3}{2} arctan (\frac{x}{2})) + \frac{1}{13} ln ∣ x - 3 ∣)

= 9 (\frac{1}{13} (2 ln x^{2} + 4 + 6 arctan (\frac{x}{2})) + \frac{9}{13} ln ∣ x - 3 ∣) - 3 (\frac{1}{13} (- \frac{3}{2} ln x^{2} + 4 + 2 arctan (\frac{x}{2})) + \frac{3}{13} ln ∣ x - 3 ∣) + 19 (\frac{1}{13} (- \frac{1}{2} ln x^{2} + 4 - \frac{3}{2} arctan (\frac{x}{2})) + \frac{1}{13} ln ∣ x - 3 ∣)

化简

9 (\frac{1}{13} (2 ln x^{2} + 4 + 6 arctan (\frac{x}{2})) + \frac{9}{13} ln ∣ x - 3 ∣) - 3 (\frac{1}{13} (- \frac{3}{2} ln x^{2} + 4 + 2 arctan (\frac{x}{2})) + \frac{3}{13} ln ∣ x - 3 ∣) + 19 (\frac{1}{13} (- \frac{1}{2} ln x^{2} + 4 - \frac{3}{2} arctan (\frac{x}{2})) + \frac{1}{13} ln ∣ x - 3 ∣) : ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2}) + 7 ln ∣ x - 3 ∣

= ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2}) + 7 ln ∣ x - 3 ∣

解答补常数

= ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2}) + 7 ln ∣ x - 3 ∣ + C

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