5.3.2

[STARTS AT 3:55]

Narrator: Use the integral test to determine whether this series converges or diverges. Well, I'm going to take f of x equal to x over 3x squared plus 1. And I'll go ahead and choose capital N equal to 1. Again, if that doesn't work, we'll find something else to use there. 

So the integral from 1 to infinity of x over 3x squared plus 1 dx. If you use u substitution here, u is 3x squared plus 1 then du equals 6x dx and 1/6 du equals x dx, that will take care of our numerator there. so that is equivalent to 1/6 du 1du over u, which is going to be a natural log. 

So that's 1/6 limit is from 1 to infinity. 1/6, valuate natural log absolute value of 3x squared plus 1. And absolute values is there, evaluated from 1 to infinity. Taking the limit as x goes to infinity, this is infinity minus something. It doesn't really matter. It's going to be infinite. Going to be infinite. So that therefore sums from 1 to infinity of n over 3n squared plus 1 diverges.