Integrate Initial Proof with airSlate SignNow

Integrate initial proof

okay in this video let's talk about this improper integral from one to infinity one over x to the p power we are going to find out the values of p so that this improper integral actually converges well to do this kind of things of course we need two things first know all integrations and second belief in our limits so let's get to work first to integrate this guy we actually have to break down into cases case one is when p is equal to 1. why because you're looking at the integral from 1 to infinity 1 over x to the first power dx and to integrate 1 over x to the first power we have to use natural log so we first do the anti-derivative which is natural log of x you don't need to put the absolute value because this is positive so is that so you don't need the absolute value anyway you can put parentheses if you would like and then you go from one to infinity and as always put infinity here and then subtract you put one there and you are looking at ln of infinity minus ln of one and as you all know when you have ln of infinity you are technically taking the limit as x goes to infinity of the ln function this right here ln is always increasing and then bounded so you actually get infinity -0 well this right here is just infinity in another word this right here actually diverges because the answer to this is not finite remember convergence means that you have a finite sum for this integral this right here diverges so we get one number out of the way now we're going to consider the rest of the numbers together here is the do case 2 when p is not equal to 1. in that case let's look at this integral as the integral from 1 to infinity let me bring the power up and we are talking about x to the negative p's power dx why because right here we can just do the reverse power rule and the reverse power rule is good when p is not 1 in this case because you put one here that's negative 1 for the power anyway i'm going to add 1 for the power and that will be my new power i would have to divide it by that so i'll put down one over and let me just put down one minus p why not huh so that's the anti-derivative and i'll write it down this is going to be 1 over 1 minus p and we multiply by x to the one minus p power okay let's just put it down like this nice water and then of course we plug in plugging right we have the one and also infinity and now we have some situation because if you put infinite to here you are talking about infinity...