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For x>0, let f(x)=int1^x loget/(1+t)dt f...

For `x>0`, let `f(x)=int_1^x log_et/(1+t)dt` find the function `f(x)+f(1/x)` and show that `f(e)+f(1/e)=1/2`

Text Solution

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`f(x)=int_1^x (log_e^t)/(t+1)dt-(1)`
`f(1/x)=int_1^x(log_e^t)/(t+1)dt`
`Let t=1/h,dt=-1/h^2dt`
`if t=1,h=1 or t=1/x,4=x`
`f(1/x)=int_1^x(log_e(1/h))/(1+1/h)(-1/h^2dx)`
`f(1/x)=int_1^x(-logh(-1))/(h+1)h dh`
`f(1/x)=int_1^x9(log_e^t)/(t(t+1))dt-(2)`
Adding 1 and 2
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