integral of (6x-12)/(x^2-8x+25)

解答

\int \frac{6 x - 12}{x ^{2} - 8 x + 25} d x

4 arctan (\frac{1}{3} (x - 4)) + 3 ln x^{2} - 8 x + 25 + C

求解步骤

\int \frac{6 x - 12}{x ^{2} - 8 x + 25} d x

使用分布积分法

= 2 arctan (\frac{1}{3} (x - 4)) (x - 2) - \int 2 arctan (\frac{1}{3} (x - 4)) d x

\int 2 arctan (\frac{1}{3} (x - 4)) d x = 2 x arctan (\frac{1}{3} (x - 4)) - 3 ln x^{2} - 8 x + 25 - 8 arctan (\frac{1}{3} (x - 4))

= 2 arctan (\frac{1}{3} (x - 4)) (x - 2) - (2 x arctan (\frac{1}{3} (x - 4)) - 3 ln x^{2} - 8 x + 25 - 8 arctan (\frac{1}{3} (x - 4)))

化简

2 arctan (\frac{1}{3} (x - 4)) (x - 2) - (2 x arctan (\frac{1}{3} (x - 4)) - 3 ln x^{2} - 8 x + 25 - 8 arctan (\frac{1}{3} (x - 4))) : 4 arctan (\frac{1}{3} (x - 4)) + 3 ln x^{2} - 8 x + 25

= 4 arctan (\frac{1}{3} (x - 4)) + 3 ln x^{2} - 8 x + 25

解答补常数

= 4 arctan (\frac{1}{3} (x - 4)) + 3 ln x^{2} - 8 x + 25 + C

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