Question
QuestionThe integralVx+1 dx is convergenceTrue 0 False
Question The integral Vx+1 dx is convergence True 0 False
Answers
Decide whether the statements are true or false. Give an explanation for your answer. The integral $\int_{0}^{\infty} \frac{1}{e^{x}+x} d x$ converges.
In this question were given that if the limit under function f x x Gourgeon infinity, ico choose there are isn't implies that it girl from Sergeant Affinity F X, the X will be convergent and this them in here will be phones One count x number will be it would can consider the f X You go to one of the X they're far on This one would say the limit on the function F Act X goes to infinity in with Iko Judah's are but the integral from surgeon For today I'm the one of the x dx They were ego to the Ellen on the absolutely executed Jews urge infinity and then we get to the infinity on then months infinities go to infinity here so doesn't obey. Divergent. It was always love in one to make it easy. So someone in minus one here it would be divergent. Here
The statement here says if the integral from zero to infinity of F. Of X. C X. And the integral from zero to infinity of G. Of X. Dx both converge, then the integral from zero to infinity of F. Of X plus G of X. Dx also converges. And this is true, right? Because we know using um our integration rules, we can actually split this up into two separate integral. So meaning the integral from zero to infinity of F. Of X plus G fx. D X could be written as the integral from zero to infinity of F. Of X. Dx plus the integral from zero to infinity G. F X. The X. So if this one is convergent and this one is also convergent and that means they converge to us. Um some numbers, some finite number. So if we add to finite numbers together, then it will give us another finite number. So therefore this is convergent.
The question is that if the limit on the function f x when x great, you zero echoed your infinity and does it implies that indigo from 0 to 1 if thanks the axe and will be divergent on Dhere system in here would be phones and we can take one context about example here, let's say we have the limit. I'm the one of Ah, thanks Ah, square the banks And then we see an extra lectures over this echo to infinity, but the integral from 0 to 1 And then we have a ah expo minus 1/2 on the X and we go to, uh, uh, ex about a skirt of X and then I'm still here. I followed under their 21 and get a quick to, uh uh, Teoh and ism in his smaller than infinity. So against one will be convergent
Yeah, we are even that if the limit of the function f x x goes to infinity ICO to one, then we get the integral from zero to infinity. If thanks the axe, it will be die approach. So it doesn't will be true because when we drawn to graph near so this will be the one here. That's I have No one will be here and now inside the functioning goes to the one some grand days and then we see this indicates exactly will be the area under the curve. And this area hand will say when it goes to infinite the when. When ex coach infinity, the function goes to one. So this area here, I mean, goes to infinite day. Therefore, that's why we're gonna divergent era.