A) \[{{\sin }^{-1}}\left( \frac{2x+5}{\sqrt{37}} \right)+c\]
B) \[{{\sinh }^{-1}}\left( \frac{2x+5}{\sqrt{37}} \right)+c\]
C) \[{{\sin }^{-1}}\left( \frac{2x+5}{37} \right)+c\]
D) None of the above
Correct Answer: A
Solution :
\[\int{\frac{dx}{\sqrt{(3-5x-{{x}^{2}})}}}\] \[=\int{\frac{dx}{\sqrt{-\left[ {{x}^{2}}+5x-3+{{\left( \frac{5}{2} \right)}^{2}}-{{\left( \frac{5}{2} \right)}^{2}} \right]}}}\] \[=\int{\frac{dx}{\sqrt{-\left[ {{\left( x+\frac{5}{2} \right)}^{2}}-\frac{37}{4} \right]}}}\] \[=\int{\frac{dx}{\sqrt{{{\left( \frac{\sqrt{37}}{2} \right)}^{2}}-{{\left( x+\frac{5}{2} \right)}^{2}}}}}\] \[={{\sin }^{-1}}\left[ \frac{\left( x+\frac{5}{2} \right)}{\left( \frac{\sqrt{37}}{2} \right)} \right]+c\] \[={{\sin }^{-1}}\left[ \frac{2x+5}{\sqrt{37}} \right]+c\]
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