A) \[{{\sin }^{-1}}(2x-3)+C\]
B) \[{{\sin }^{-1}}(2x+5)+C\]
C) \[{{\sin }^{-1}}(3-2x)+C\]
D) \[{{\sin }^{-1}}(5-2x)+C\]
Correct Answer: A
Solution :
Let \[I=\int_{{}}^{{}}{\frac{dx}{\sqrt{(1-x)(x-2)}}}\] \[=\int_{{}}^{{}}{\frac{dx}{-{{x}^{2}}+3x-2}}=\int_{{}}^{{}}{\frac{dx}{\sqrt{\frac{1}{4}-{{\left( x-\frac{3}{2} \right)}^{2}}}}}\] \[={{\sin }^{-1}}\left( \frac{\left( x-\frac{3}{2} \right)}{\frac{1}{2}} \right)+C\] \[={{\sin }^{-1}}(2x-3)+C\]
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