Definite Integration Question 493
Question: The value of $ \int_2^{3}{\frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}}dx $ is
[IIT 1994; Kurukshetra CEE 1998]
Options:
A) 1
B) 0
C) $ -1 $
D) $ \frac{1}{2} $
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Answer:
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} $ .
BETA
