If f and g are continuous functions, then ln1λ∫lnλf(x24)[f(x)−f(−x)]g(x24)[g(x)+g(−x)]dx,λ>0, is
A
dependent on λ only
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B
a non-zero constant
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C
zero
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D
dependent on f and g only
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Solution
The correct option is C zero Let us consider H(x)=f(x24)[f(x)−f(−x)]g(x24)[g(x)+g(−x)] ⇒H(−x)=f(x24)[f(−x)−f(x)]g(x24)[g(−x)+g(x)] ⇒H(−x)=−f(x24)[f(x)−f(−x)]g(x24)[g(x)+g(−x)] ⇒H(−x)=−H(x) So, it is an odd function.