What is the integral of ((8x-11))/(sqrt(5+2x-x^2)) ?

The integral of ((8x-11))/(sqrt(5+2x-x^2)) is -8sqrt(-x^2+2x+5)-3arcsin(1/(sqrt(6))(x-1))+C

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

\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

Solution steps

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

Expand

\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

Apply the Sum Rule:

\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))

Simplify

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))

Add a constant to the solution

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

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What is the integral of ((8x-11))/(sqrt(5+2x-x^2)) ?
The integral of ((8x-11))/(sqrt(5+2x-x^2)) is -8sqrt(-x^2+2x+5)-3arcsin(1/(sqrt(6))(x-1))+C