integral of (3x^2+8x-12)/(x^3+7x+12x)

解答

\int \frac{3 x ^{2} + 8 x - 12}{x ^{3} + 7 x + 12 x} d x

\frac{69}{38} ln x^{2} + 19 - \frac{12}{19} ln ∣ x ∣ + \frac{8}{19} arctan (\frac{x}{19}) + C

求解步骤

\int \frac{3 x ^{2} + 8 x - 12}{x ^{3} + 7 x + 12 x} d x

乘开

\frac{3 x ^{2} + 8 x - 12}{x ^{3} + 7 x + 12 x} : \frac{3 x ^{2}}{x ^{3} + 7 x + 12 x} + \frac{8 x}{x ^{3} + 7 x + 12 x} - \frac{12}{x ^{3} + 7 x + 12 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} + 7 x + 12 x} d x + \int \frac{8 x}{x ^{3} + 7 x + 12 x} d x - \int \frac{12}{x ^{3} + 7 x + 12 x} d x

\int \frac{3 x ^{2}}{x ^{3} + 7 x + 12 x} d x = \frac{3}{2} ln x^{2} + 19

\int \frac{8 x}{x ^{3} + 7 x + 12 x} d x = \frac{8}{19} arctan (\frac{x}{19})

\int \frac{12}{x ^{3} + 7 x + 12 x} d x = 12 (\frac{1}{19} ln ∣ x ∣ - \frac{1}{38} ln x^{2} + 19)

= \frac{3}{2} ln x^{2} + 19 + \frac{8}{19} arctan (\frac{x}{19}) - 12 (\frac{1}{19} ln ∣ x ∣ - \frac{1}{38} ln x^{2} + 19)

化简

\frac{3}{2} ln x^{2} + 19 + \frac{8}{19} arctan (\frac{x}{19}) - 12 (\frac{1}{19} ln ∣ x ∣ - \frac{1}{38} ln x^{2} + 19) : \frac{69}{38} ln x^{2} + 19 - \frac{12}{19} ln ∣ x ∣ + \frac{8}{19} arctan (\frac{x}{19})

= \frac{69}{38} ln x^{2} + 19 - \frac{12}{19} ln ∣ x ∣ + \frac{8}{19} arctan (\frac{x}{19})

解答补常数

= \frac{69}{38} ln x^{2} + 19 - \frac{12}{19} ln ∣ x ∣ + \frac{8}{19} arctan (\frac{x}{19}) + C

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