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Help prove `int_0^1 x^n/(5x+7)dx >= int_0^1 x^(n+1)/(5x+7) dx`  ?

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You need to prove that `int_0^1 x^n/(5x+7)dx >= int_0^1 (x^(n+1))/(5x+7)dx`  such that:

`x in [0,1] => x^n > x^(n+1)`

Since `5x + 7 > 0`  if `x in [0,1], ` then `x^n/(5x+7) > (x^(n+1))/(5x+7), ` for`x in [0,1].`

You need to integrate the inequality x^n/(5x+7) > (x^(n+1))/(5x+7) such that:

`int_0^1 x^n/(5x+7)dx > int_0^1 (x^(n+1))/(5x+7)dx `

Hence, using only the fact that the power function decreases over [0,1] and the expression `5x+7`  is positive for `x in [0,1], ` yields that`int_0^1 x^n/(5x+7)dx > int_0^1 (x^(n+1))/(5x+7)dx.`

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greenspo | Student

integral (up1 down 0)x^n/5x+7 > equal integral(up 1 down 0) x^n+1/5x+7