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