# Prove $1 \lt \int_0^1 \frac{1+x^{30}}{1+x^{60}} \ \mathrm{d}x \lt 1 + \frac{1}{30}$ [duplicate]

How can I prove the estimate $$1 \lt \int_0^1 \frac{1+x^{30}}{1+x^{60}} \ \mathrm{d}x \lt 1 + \frac{1}{30}?$$ Of course, the lower bound is pretty obvious. I realize this looks like a homework problem, but it's not (it's actually an old qualifying exam question). I think it is possible to use contour integration in the complex plane to get an exact expression for the integral, but this is pretty complicated and hopefully there is a way to estimate the integral without evaluating it.

## marked as duplicate by David Mitra, Argon, user17762, Justin Campbell, DidAug 4 '12 at 21:14

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• See here. – David Mitra Aug 4 '12 at 19:58
• Oops, I didn't see the duplicate. I'm voting to close as well. – Justin Campbell Aug 4 '12 at 20:35

## 1 Answer

$$1 < \dfrac{1+x^{30}}{1+x^{60}} < 1+x^{30} \,\,\,\,\, \forall x \in (0,1)$$

• I think this is the best way one may think of. (+1) – user 1357113 Aug 9 '12 at 13:23