A) \[4\]
B) \[2\]
C) \[1\]
D) \[1/2\]
Correct Answer: B
Solution :
\[I=\int_{4}^{8}{\frac{\sqrt{x}}{\sqrt{x}+\sqrt{12-x}}}\,\,dx\] ?.(i) \[\because \] \[\int_{a}^{b}{f(x)\,dx=\int_{a}^{b}{ff(a+b-x)dx}}\] \[I=\int_{4}^{8}{\frac{\sqrt{(8+4-x)}}{\sqrt{8+4-x}+\sqrt{12-(8+4-x)}}}\,dx\] \[I=\int_{4}^{8}{\frac{\sqrt{12-x}}{\sqrt{12-x+\sqrt{x}}}\,}\,dx\] ?.(ii) On adding Eqs. (i) and (ii), we get \[2I=\int_{4}^{8}{\frac{\sqrt{x}+\sqrt{12-x}}{\sqrt{x}+\sqrt{12-x}}}\,\,dx=\int_{4}^{8}{1\,\,dx}\] \[2I=[x]_{4}^{8}\,=\,\,[8-4]\] \[\Rightarrow \] \[2I=4\] \[\Rightarrow \] \[I=2\]
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