A) 1
B) 0
C) \[-1\]
D) \[\frac{1}{2}\]
Correct Answer: D
Solution :
\[I=\int_{2}^{3}{\frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}dx}\] ?..(i) Using the property \[I=\int_{a}^{b}{f(x)dx=\int_{a}^{b}{f(a+b-x)}dx}\] i.e., change in\[x=(2+3-x)=5-x\]or \[dx=-dx\] \[\therefore I=\int_{3}^{2}{\frac{\sqrt{5-x}}{\sqrt{x}+\sqrt{5-x}}}(-dx)\]\[=\int_{2}^{3}{\frac{\sqrt{5-x}}{\sqrt{5-x}+\sqrt{x}}dx}\] ?..(ii) Adding (i) and (ii), \[2I=\int_{2}^{3}{\frac{\sqrt{x}+\sqrt{5-x}}{\sqrt{5-x}+\sqrt{x}}dx=\int_{2}^{3}{1dx}}\] \[=[x]_{2}^{3}=3-2=1\Rightarrow I=\frac{1}{2}\].
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