Strange Integration

Try Integration By Parts

[tex]\int[/tex] u dv = uv - [tex]\int[/tex] v du

 

substitute u = sqr(x), then use partial fractions

 

Oh, great. So Wolfram Alpha will not only tell you the answer to your homework, it will tell you what intermediate steps to write down. That's nasty.

 

You can't really 'integrate' using the comparison method, but you can show the integral exists and find a bound on it. If that's all you want to do, it's a lot easier than actually doing the integration. To deal with the interval [0,1] you had the right idea when you looked at the graph and observed it's approximately between 0 and 1/2. You could just compute the upper bound by finding the derivative, looking for critical points etc. If you do you'll find it's actually a bit larger than 1/2.

 

Computing the integral is a piece of cake using contour integration. In fact, it is so easy you can do it in your head. It is:

[tex]\frac{\pi}{\sqrt{2}}[/tex]

 

The maximum on [0,1] is attained at x=1/sqrt(3) making the value of the integral over [0,1] less than f(1/sqrt(3)) where f(x)=sqrt(x)/(x^2+1). That's what you meant to say, right? You could also do a rougher comparison by noting sqrt(x)/(1+x^2)<=1/1, the max of the numerator over the min of the denominator.

 

Sure you could. If you wanted to narrow it more you could cut [0,1] into subintervals, and also split [1,infinity) into subintervals. Depends on what you want to do. You could also with some work integrate it exactly using either substitution and partial fraction as waht suggested or a contour integration as Count Iblis suggested. If you haven't done contour integrals before I wouldn't suggest starting with this one. And waht's technique is kind of a long haul too, but doesn't require complex analysis. Like I say, depends on what kind of answer you are happy with.

 

You're welcome. Wolfram Alpha seems pretty dangerous to me. I can imagine some people might have a hard time resisting the urge to peek at the answer before they even try to solve the problem. If the contents of WA's 'Show steps' were posted on the Forum as the response to a question the moderators would delete it.