A) \[1\]
B) \[3\]
C) \[2\]
D) \[0\]
Correct Answer: A
Solution :
Let \[I=\int_{1}^{3}{\frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}}}dx\] ??(i) \[\Rightarrow \] \[I=\int_{1}^{3}{\frac{\sqrt{4-(4-x)}}{\sqrt{4-x}+\sqrt{4-(4-x)}}}dx\] \[[\,\because \,\,\int_{a}^{b}{f(x)\,dx=\int_{a}^{b}{f(a+b-x)dx]}}\] \[\Rightarrow \] \[I=\int_{1}^{3}{\frac{\sqrt{x}}{\sqrt{4-x}+\sqrt{x}}}dx\] ??(ii) On adding Eqs. (i) and (ii), we get \[2I=\int_{1}^{3}{1\,dx=[x]_{1}^{3}}\] \[\Rightarrow \] \[I=\frac{2}{2}=1\]
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