integral of (6x^3+2x^2-47x-9)/(x^2-9)

解答

\int \frac{6 x ^{3} + 2 x ^{2} - 47 x - 9}{x ^{2} - 9} d x

3 x^{2} + 2 x - \frac{3}{2} ln \frac{x}{3} + 1 + \frac{3}{2} ln \frac{x}{3} - 1 + \frac{7}{2} ln x^{2} - 9 - 27 + C

求解步骤

\int \frac{6 x ^{3} + 2 x ^{2} - 47 x - 9}{x ^{2} - 9} d x

乘开

\frac{6 x ^{3} + 2 x ^{2} - 47 x - 9}{x ^{2} - 9} : \frac{6 x ^{3}}{x ^{2} - 9} + \frac{2 x ^{2}}{x ^{2} - 9} - \frac{47 x}{x ^{2} - 9} - \frac{9}{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{6 x ^{3}}{x ^{2} - 9} d x + \int \frac{2 x ^{2}}{x ^{2} - 9} d x - \int \frac{47 x}{x ^{2} - 9} d x - \int \frac{9}{x ^{2} - 9} d x

\int \frac{6 x ^{3}}{x ^{2} - 9} d x = 3 (x^{2} - 9 + 9 ln x^{2} - 9)

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

\int \frac{47 x}{x ^{2} - 9} d x = \frac{47}{2} ln x^{2} - 9

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

= 3 (x^{2} - 9 + 9 ln x^{2} - 9) - 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) - \frac{47}{2} ln x^{2} - 9 - (- 3 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1))

化简

3 (x^{2} - 9 + 9 ln x^{2} - 9) - 6 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1 - \frac{x}{3}) - \frac{47}{2} ln x^{2} - 9 - (- 3 (\frac{1}{2} ln \frac{x}{3} + 1 - \frac{1}{2} ln \frac{x}{3} - 1)) : 3 x^{2} + 2 x - \frac{3}{2} ln \frac{x}{3} + 1 + \frac{3}{2} ln \frac{x}{3} - 1 + \frac{7}{2} ln x^{2} - 9 - 27

= 3 x^{2} + 2 x - \frac{3}{2} ln \frac{x}{3} + 1 + \frac{3}{2} ln \frac{x}{3} - 1 + \frac{7}{2} ln x^{2} - 9 - 27

解答补常数

= 3 x^{2} + 2 x - \frac{3}{2} ln \frac{x}{3} + 1 + \frac{3}{2} ln \frac{x}{3} - 1 + \frac{7}{2} ln x^{2} - 9 - 27 + C

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\frac{d}{d x} (\frac{1}{2} ln^{2} (x))

d er i v a t i v e - 7 csc (5 x) e^{3 x}