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If a continuous function f on [0,a] sati...

If a continuous function `f` on `[0,a]` satisfies `f(x)f(a-x)=1,agt0`, then find the value of `int_(0)^(a)(dx)/(1+f(x))`.

Text Solution

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The correct Answer is:
`a//2`
`f(x)f(a-x)=1` or `f(a-x)=1/(f(x))`
Now, `I=int_(0)^(a)(dx)/(1+f(x))=int_(0)^(a)(dx)/(1+f(a-x))`
`=int_(0)^(a)(dx)/(1+1/(f(x)))`
`=int_(0)^(a)(f(x)dx)/(1+f(x))`
`:. 2I=int_(0)^(a)(1+f(x))/(1+f(x))dx=a` or `I=a//2`
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