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Odd Function

If I=∫8x-11√5...

Question

If $I = \int \frac{8 x - 11}{\sqrt{5 + 2 x - x^{2}}} d x = p \sqrt{5 + 2 x - x^{2}} + q {sin}^{- 1} (\frac{x - 1}{\sqrt{6}}) + C,$ then the value of $| p + q |$ ; $(p, q \in R)$ is
(where $C$ is integration constant)

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Solution

$I = \int \frac{8 x - 11}{\sqrt{5 + 2 x - x^{2}}} d x = \int \frac{- 4 (2 - 2 x) - 3}{\sqrt{5 + 2 x - x^{2}}} d x = - 4 \int \frac{(2 - 2 x)}{\sqrt{5 + 2 x - x^{2}}} d x - 3 \int \frac{d x}{\sqrt{6 - (x - 1)^{2}}} = - 8 \sqrt{5 + 2 x - x^{2}} - 3 {sin}^{- 1} (\frac{x - 1}{\sqrt{6}}) + C \Rightarrow p = - 8, q = - 3 ∴ | p + q | = | - 8 - 3 | = 11$

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