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# Sketch the region and find its area (if the area is finite).$S = \{ (x, y) \mid x \ge 1, 0 \le y \le \frac{1}{(x^3 + x)} \}$

## $\frac{1}{2} \ln 2$

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### Video Transcript

travel always sketched region and find its area. Look at this graph. Here, we need to find his area of the shady part. This part we can use into girl from Want to infinity one over X two Q A sax, Jax. Everything now area with this part for this improper integral A definition. This is the cultures that limit a cause to infinity. To go from one toe a hell's function. One over. Ax to kill class ax the ax. Now compute this definite integral first. But this staff in the euro look at dysfunction one over as to kill with class acts one over Act two Q Our sides, this is Echo two. Want over X times X squared plus one. We can revive his function. You got tio some number A over X US B X. I see two x plus one and then compute to constant number A B C back empire the coefficients off both sides a competition. This is equal to one over. Max. Negative. Ax over. Explain us one. Now, Stephanie, the integral is equal to a girl from want to, eh? One over ice. Yeah, Thanks. My minus into girl from Want eh? Act's over. Explain plus one. Yes, On this part this is Ego to Ellen. Us from one way says the sequel to L. A For this part. Yeah. We use you substitution first. That view is we go to X Claire. Ten u Is ICO too Two times axed. Yes, this is Echo two into girl. From what, Too square is the one half you over. You plus one. This's equal to one half hour. You plus one from one to a square. This is equal to one. Have you? Yeah. And a square Paswan minus one half. How and Teo. Now, when a minus. Alan. Now, when they minus one off? Yeah, and a square house. Juan US one, Huh? No. Two. This is what the culture is. This one now No cavity disfunction. So we can write dysfunction on a were minus one half times in a square. Paswan, this is going to intruder. A squire Us Want class one half now and two now what a goes to infinity. A over motive A squared plus one cost one. I'll end Teo How end a over a beautif a square plus one goes to zero. So this thiss result goes to one half No. Two two. Since the area of this region is equal to one half, I want to Yeah, Here. You need to notice that we use this two functions out in some a minus on two B is equal to L. A and A over B on DH. One half out in some axe, this's equal Teo and act two one half Or you can write system out root of ax thiss postal them our identities.

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