Integral Calculation - Urgent

[joreto_co]

Hello,

I need to show that an integral from minus infinity to infinity is smaller than infinity. All the expressions that are integrated are positive. What should I do? Can you recommend me some theorems that I can use? Thank you in advance!

 

[galactus]

It would help if you would post your problem. It's difficult to say without knowing what you're faced with.

 

[joreto_co]

here are the two inequalities I have to prove (the second in both directions)

 

[cadamcross]

Thoughts on the first one:

I'll assume x>= 0. Note that the function is not defined for x=0, but we can say it equals the limit as x approaches 0. It may be prudent to consider the domain in two separate cases: 0 <= x <= 1 and x > 1. You can show that the function is bounded on 0 <= x <= 1. Outside this interval, the function is positive, and I recommend bounding it above by a function with x substituted for log(x). This may not be the simplest or most elegant. It's just what comes to mind.