I want ask how to start to tackle the problems of inequalities involving integrals:

#1 Prove that

#2 Prove that

#3 Prove that

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- December 15th 2011, 05:42 AMmaoroproving integral inequalities
I want ask how to start to tackle the problems of inequalities involving integrals:

#1 Prove that

#2 Prove that

#3 Prove that

- December 15th 2011, 08:33 AMSironRe: proving integral inequalities
You can compute the first integral by using integration by parts.

- December 15th 2011, 09:23 AMmaoroRe: proving integral inequalities
- December 15th 2011, 10:04 AMSironRe: proving integral inequalities
Yes, that's correct, you can simplify it to:

and moreover:

Therefore you have proved the inequality. - December 15th 2011, 10:10 AMmaoroRe: proving integral inequalities
- December 15th 2011, 10:15 AMSironRe: proving integral inequalities
(2)

(1)

And - December 15th 2011, 10:20 AMmaoroRe: proving integral inequalities
Then is the process the same for #2?

It's quite complicated when evaluating the definite integral - December 15th 2011, 10:33 AMSironRe: proving integral inequalities
No, It won't be the same, because indeed as you note they're very complex to evaluate and moreover you can't express in term of standard mathematical functions, similar for the second one.

I would do something with the fact that you can evaluate: