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Question

If f(x) and g(x) are continuous functions, the ln1/λlnλf(x2/4)[f(x)f(x)]g(x2/4)[g(x)+g(x)]dx is

A
dependent on λ
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B
a non-zero constant
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C
zero
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D
None of these
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Solution

The correct options are
A dependent on λ
C zero
I=log1λlogλf(x2/4)[f(x)f(x)]g(x2/4)[g(x)+g(x)]dx

=logλlogλf(x2/4)[f(x)f(x)]g(x2/4)[g(x)+g(x)]=0
Let h(x)=f(x2/4)[f(x)f(x)]g(x2/4)[g(x)+g(x)]

h(x)=f((x)2/4)[f(x)f(x)]g((x)2/4)[g(x)+g(x)]

h(x)=f(x2/4)[f(x)f(x)]g(x2/4)[g(x)+g(x)]

h(x)=f(x2/4)[f(x)f(x)]g(x2/4)[g(x)+g(x)]

h(x)=h(x)

Thus, the function is odd.
I=0

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