A) \[(-2,-1)\]
B) \[(2,-1)\]
C) \[(-2,1)\]
D) \[(2,1)\]
Correct Answer: A
Solution :
Note that \[7-6x-{{x}^{2}}=16-{{(x+3)}^{2}}\] and\[\frac{d}{dx}(7-6x-{{x}^{2}})=-2x-6\] So, we have \[\begin{align} & \int_{{}}^{{}}{\frac{2x+5}{\sqrt{7-6x-{{x}^{2}}}}dx=\int_{{}}^{{}}{\frac{2x+6}{\sqrt{7-6x-{{x}^{2}}}}dx}} \\ & -\int_{{}}^{{}}{\frac{1}{\sqrt{16-{{(x+3)}^{2}}}}dx} \\ \end{align}\] \[=-2\sqrt{7-6x-{{x}^{2}}}-{{\sin }^{-1}}(\frac{x+3}{4})+C\] So, we have \[A=-2,B=-1\]. Thus option A is the correct answer.
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