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Prove that int(0)^(1)(dx)/(1+x^(n))gt1-(...

Prove that `int_(0)^(1)(dx)/(1+x^(n))gt1-(1)/(n)"for n"inN`

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`x^(n-1)-x^(n)+x^(2n-1)`
`" "=x^(n-1)(1-x)+x^(2n-1) gt 0 " for "x in (0,1)`
`rArr" "int_(0)^(1)(x^(n-1)-x^(n)+x^(2n-1))/(1+x^n)dxgt 0`
`rArr" "int_(0)^(1){(1)/(1+x^(n))-(1-x^(n-1))}dxgt0`
`rArr" "int_(0)^(1)(dx)/(1+x^(n))gt int_(0)^(1)(1-x^(n-1))dx`
`rArr" "int_(0)^(1)(dx)/(1+x^(n))gt1-(1)/(n),AA n in N`
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