integral of (-2x+11)/(x^2-8x+25)

解答

\int \frac{- 2 x + 11}{x ^{2} - 8 x + 25} d x

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

求解步骤

\int \frac{- 2 x + 11}{x ^{2} - 8 x + 25} d x

乘开

\frac{- 2 x + 11}{x ^{2} - 8 x + 25} : - \frac{2 x}{x ^{2} - 8 x + 25} + \frac{11}{x ^{2} - 8 x + 25}

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}{x ^{2} - 8 x + 25} d x + \int \frac{11}{x ^{2} - 8 x + 25} d x

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

\int \frac{11}{x ^{2} - 8 x + 25} d x = \frac{11}{3} arctan (\frac{1}{3} (x - 4))

= - (ln x^{2} - 8 x + 25 + \frac{8}{3} arctan (\frac{1}{3} (x - 4))) + \frac{11}{3} arctan (\frac{1}{3} (x - 4))

化简

- (ln x^{2} - 8 x + 25 + \frac{8}{3} arctan (\frac{1}{3} (x - 4))) + \frac{11}{3} arctan (\frac{1}{3} (x - 4)) : - ln x^{2} - 8 x + 25 + arctan (\frac{1}{3} (x - 4))

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

解答补常数

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

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