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Evaluate the integrals that converge.$$\int_{0}^{+\infty} \frac{1}{x^{2}} d x$$

Divergent

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

PRINCIPLES OF INTEGRAL EVALUATION

Section 8

Improper Integrals

Integrals

Integration

Integration Techniques

Trig Integrals

Trig Substitution

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we have the integral going from zero to infinity of well, one over X squared DX. So both of the endpoints here, right? We have a problem if we have while the lower end point is zero why it gives us an improper integral because we have in the nominator and also the upper end point was infinite, improper, integral on both of our endpoints. So therefore we can take well is the limit Um, twice here, right, This is equal to the limit as well. Um, let's say be approaches zero of the integral going from be up to, let's say, one of this option D x over x squared and then plus the limit as well be approaches infinity Oops, infinity of then while going from one, um to of one to be what I will be once I can do something Let's see, we just maybe maybe k so the arm limit as people to zero of the integral going from beetle one of d x or X squared, plus the limit. Thats a kay approaches that infinity of then going from where we stopped one to you invented of the same function here D x over X squared. Okay, so then Well, evaluating this mineral we get this is equal to the limit as be approaches zero off. Well, negative. Um, won over. Axe evaluated from b toe one plus the limit as K approaches approaches infinity of well, negative one over. X waiting from 12 Okay, okay, so we get this is equal to I'm the limit as he approaches zero where we get we get negative one. Just negative 1/1 plus one over b and then plus the limit as que approaches infinity off while negative one. Okay. Plus one. Over. Okay, Well, native of one is just negative one, right. Plus one over. Be taking the limit here. We just get well. This is equal to the native. One can come outside. The limit is going to be equal to the limit as beef approaches. Zero off just well. One Overbey. And then we have wild minus one. Right. And then here was K approaches infinity. Negative one over infinity. That's close to zero plus one. So this is just the limit as being put zero, what will be minus one and then this limit, just like I was the one we have plus one. Well, one over. Be right. It goes to infinity. So infinity minus one plus one. That's just mean minus one plus one is zero. If you want to think about this doesn't matter, Right? This is just equal to the limit as the approaches zero off, one over be well, that's equal to infinity. Right? So therefore the given integral integral is guy Urgent, diverted. Get a We don't get a finite answer for a limit. Begin infinity! So therefore the improper integral we have is divergent.

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