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Problem 71 Hard Difficulty

Determine how large the number a has to be so that
$$ \int_a^\infty \frac{1}{x^2 + 1}\ dx < 0.001 $$

Answer

$I<0.001 \Rightarrow \frac{\pi}{2}-\tan ^{-1} a<0.001 \Rightarrow \tan ^{-1} a>\frac{\pi}{2}-0.001 \Rightarrow a>\tan \left(\frac{\pi}{2}-0.001\right) \approx 1000$

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Video Transcript

Hello. Welcome to this lesson in this lesson. We're looking at the value of a forge, the whole of line to grow. It's less than 0.1 So let's begin by looking at this bad. Cool. Okay. Mhm A sovereign that you let X equals two. And why? Okay, So that the X would be called to It's squared. Why? Why part? All right. So it means are in place of the DX who puts a squared. Why do I? So let's go on. So a to infinity in place of the why we could Sex square Why do I then we now have? Yeah. Don't square y plus one. Thank you, sis. A than Infinity says spread. Why, uh turn squared, wipe. Last one, geisha six squared Y Wilson s T y. So this will cross out that. So now we have then to grow a two infinity, do y Oh, we can also replace the wet the limit as t approaches infinity A t. Why? Um Mhm. So this is where we have? Yeah. So we have a soul body sick or two. The limit s t approaches Infinity. Yeah. If we integrated whole salary of Why than 18? Okay, now we saw that X was recalled to turn y okay, so we can be placed back. Um oh, we can replace. Uh, y okay, So it means that why is equal to turn, invest or the act on of X? So the whole thing right now it costs the limits, s t approaches Infinity of the trend invests both banks. Yeah, a two t So that is the limits again. Mhm. Mhm. Yeah. So turn invest of t than minus and invest P. Yeah. Now, the limit of t the limits as he approaches infinity of tanning beds of tea gives us a pie on to us. The number increases indefinitely. The turning vests or the Acton of that number becomes pie onto, so this can be Whoa! Mhm! I on two minus turn Invest of a. Okay, But we have been told that the whole thing here is is less than 0.1 So taking it from here, who have further for less ad the turn invests all the Acton to both sides. Yeah. Yeah. Then they attempt to both sides. Okay, start this and that will go away. Yes. Yeah, Okay. You know, the next thing to do is that we subtract the open. There's no one from both sides. Yeah. Come on called. Yeah. Okay. Okay. Yeah. Any form? Mhm. Okay. So that we have mhm. Uh huh. It's less than Afghan bigger. Okay, so here we would have 1.56 nine. Or probably instead of thinking that the whole thing on two Okay, we can take the time on both sides so that this becomes a so he should be greater than the ton of That's okay. And the whole of this. Oh, the whole of this gives us. Yeah. 1000. Mhm. Really? Okay, So it means that a should be greater than 1000. Yeah. All right. If we bury the whole of this 1000. Almost. Okay, so thank you. Thanks for a time. This is the end of the lesson.