Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I need help proving this inequality.

$$\int_{1}^{2} \left(\frac{x+1}{2} \right)^{x+1} \ dx < \int_{1}^{2} x^x \ dx$$

Normally I would define $f(x) = \left(\displaystyle\frac{x+1}{2} \right)^{x+1}$ and $g(x) = x^x$ and try to compare those two.
Any suggestions?

share|cite|improve this question
Evaluate each integrals separately. For f(x) and g(x), evaluate using numerical methods. – Neigyl Noval Jan 23 '11 at 11:49
up vote 6 down vote accepted

HINT: In $[1,2]$ we have that $x^x > (\frac{x+1}{2})^{x+1}$. Try to show this by finding the minimum of $h(x) = x\log(x)-(x+1)\log(\frac{x+1}{2})$.

share|cite|improve this answer
oh good point! thanks – andrei Jan 24 '11 at 13:34

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.