integral of (8x-11)/(sqrt(5+2x-x^2))

解答

\int \frac{8 x - 11}{5 + 2 x - x ^{2}} d x

- 8 - x^{2} + 2 x + 5 - 3 arcsin (\frac{1}{6} (x - 1)) + C

求解步骤

\int \frac{8 x - 11}{5 + 2 x - x ^{2}} d x

乘开

\frac{8 x - 11}{5 + 2 x - x ^{2}} : \frac{8 x}{5 + 2 x - x ^{2}} - \frac{11}{5 + 2 x - x ^{2}}

= \int \frac{8 x}{5 + 2 x - x ^{2}} - \frac{11}{5 + 2 x - x ^{2}} d 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{8 x}{5 + 2 x - x ^{2}} d x - \int \frac{11}{5 + 2 x - x ^{2}} d x

\int \frac{8 x}{5 + 2 x - x ^{2}} d x = 8 (- - x^{2} + 2 x + 5 + arcsin (\frac{1}{6} (x - 1)))

\int \frac{11}{5 + 2 x - x ^{2}} d x = 11 arcsin (\frac{1}{6} (x - 1))

= 8 (- - x^{2} + 2 x + 5 + arcsin (\frac{1}{6} (x - 1))) - 11 arcsin (\frac{1}{6} (x - 1))

化简

8 (- - x^{2} + 2 x + 5 + arcsin (\frac{1}{6} (x - 1))) - 11 arcsin (\frac{1}{6} (x - 1)) : - 8 - x^{2} + 2 x + 5 - 3 arcsin (\frac{1}{6} (x - 1))

= - 8 - x^{2} + 2 x + 5 - 3 arcsin (\frac{1}{6} (x - 1))

解答补常数

= - 8 - x^{2} + 2 x + 5 - 3 arcsin (\frac{1}{6} (x - 1)) + C

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