1The value of the integral J Jo xy Vx2 +y2dydx is2(213 _ 1) 15TrueFails...

$1-82$ Evaluate the integral.
$$\int_{-1}^{1} \frac{e^{\arctan y}}{1+y^{2}} d y$$

For this problem we are asked to evaluate the integral from negative 1 to 1 of each of the power of arc tanne. Why divided by one plus y squared? So to begin we have what we can do Is make the substitution that U equals 10 inverse of why? Which would then mean that D. U. Is going to be won over Y squared plus one. And we need to take 10 inverse of one and negative one to change the end points there. So arc tanne of positive one is pi by four. So our upper boundary is going to be pi by four and arc 10 of negative one is negative pi by four. So we are going to have the integral from negative pi by 42. Pi by four of just me. Yeah it's going to be just E to the power of you now because the one over y squared plus one gets canceled out. So integral of eat. Our view is just eat our view and we're evaluating that from negative pi by four up to pi by four. So we'll have e to the power of pi by four minus one over E. To the power of pi by four. So we can simplify that down a little bit if we wanted to and write this as E to the power of pie by two minus one divided by E. To the power of pi by four

Been in a great 1/1 plus x plus. Why over this region and the region is X goes from 1 to 3. And why goes from 1 to 2? So we'll do. 122123 1/1 plus x plus. Why 123 was X So DX on the inside. D y on the outside. Okay, it's ah, log of rhythm. So I'm gonna go ahead and do this substitution so I can make sure I get it right. If you we're gonna make you b one plus x plus y, then do you is DX. And if x is one, then you is two plus y. If X is three, then you is four plus y. So this is theater girl 1 to 2 two plus y four plus y one over you. Do you? Do you? Why? So it's 1 to 2 Ellen of you from two plus y to four plus y Do you wife? So it's going to to Elena four plus y minus Yellen of two plus y. Do you worry? Okay, now we're gonna integrate, um, the l n. And if we made this, you then do you would just be D y. So we don't even have to write that down formula. Remember, for the integral of Yellen is X Ln x minus x. Let's write that one down Ln x t x equals ex Elin X minus X plus e So here X is four plus wise. So we get four plus y Ellen of four plus Why minus four plus why minus now we're going to integrate this one. And for this one, we get two plus. Why Ellen of two plus Why minus to plus Why. Oops, That's for 1 to 2 So six Ellen six minus six minus for Ellen for minus four minus five. Ellen five minus five minus three l N. Three minus three So six Ellen six minus four. Ellen four minus five. Ellen five plus three. Ellen three minus six plus four plus five minus minus minus minus three. Good. So six. Ellen six minus five. Ellen five minus four. Ellen four plus three Ellen three. I was nine plus 90 that's my answer there. We might be able to fix it up by combining these, but it's not gonna look that much better if we dio, so I'm gonna leave it like that

Okay if we make a substitution here W for you minus V. S. And D. W would be D. You. So I don't even have to write it can just integrate I get u minus free to the sixth over six from 0 to 1 T. V. So 16011 minus V. To the sixth minus minus V. To the sixth. D. Re Okay so if you is one minus V then D. You would be minus dvs or minus and a minus. And then the same thing for that too. So we get -1 6th You to the 6th do you? So 1 -3. To the 7th over seven minus minus V. To the seventh over seven from 01. So -1/42 0 to the 7th -1 to the 7th -1 to the 7th 0 to the 7th. So -1 over 420 minus negative one minus one zero.

The core group of I minus five divided by bias. People must will simply by that this expressive so we can apply that formula for that big is so. First of all, simplify this one You can I limit one comport We wonder why. Why? It's where minus why may Why yes, Where include beauty Any route under of I'm in the middle of the power of why anyone Britain and the power off white area schools didn't get an early one local. Why get the power minus TD by? Yeah, you get one bite white Yeah, it right one day Why Egypt will who like a power minus one. Here the power is minus one so we can't apply the power of the The Parenti is a vaginal power off India s elite except in India's X step out in place of on divided by impossible plus C where any Oriel will accept an ethical duke minus so it can't apply pay And this formula one way of life here. So what did they think? It's a fair point. Maybe so in Ferguson up one my ex, it'll be it people to Ellen x 11 x plus c So they think it's enough One by accident. In the next, More doctors. So, like this formal. And here they get no power off. Why did parties minus one big divided right minus one day and the end Egotism of one by wine It Why LTD one go for not simply by men in blood in the limb. Here you get minus to buy. Why keep our one bacon minus Ellen? More duct way limited. I want to now liking that. Leave it here. Then you will get a parliament. My left toe divided by world on the power play to minus Ellen Who? What is a poignant it Soon get back that clear. Mine has no political lower limits. So we'll get manner to buy one. Get about one to minus Ellen. But they look a little on it. Do here, you idiot! My nephew. Way video. Forgive our way, please To minus four my due minus. Do not simply pay more. Minus two by two each minus one Right here. Not required to break it right. Minus one minus a lid minus minus less. Who so we'll get a minus one plus two Each one. So the final inside these one minus Thank you