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Evaluale the inlegral by inlcrpa ung it in Ierms 0 ared (Hint: Sketch the rgion indicaled by the nlegrind und imils ol integration . You may ako wunt (0 consider ...

Question

Evaluale the inlegral by inlcrpa ung it in Ierms 0 ared (Hint: Sketch the rgion indicaled by the nlegrind und imils ol integration . You may ako wunt (0 consider the fuct that the inlegral o # sum iS te sum ot Lhe integrals provicked tEcy exist )(a) E 5rMdr()5"(+V1G P)r

Evaluale the inlegral by inlcrpa ung it in Ierms 0 ared (Hint: Sketch the rgion indicaled by the nlegrind und imils ol integration . You may ako wunt (0 consider the fuct that the inlegral o # sum iS te sum ot Lhe integrals provicked tEcy exist ) (a) E 5rMdr ()5"(+V1G P)r



Answers

In Exercises 39–48, evaluate the definite integral. Use a graphing utility to confirm your result.
$$
\int_{0}^{1} e^{x} \sin x d x
$$

In this video we'll show how to integrate this integrated integral. So we have our function here, eat at the bar explains why. So there is a function of X and Y. Then we're giving our order dx dy And also given 1° from 0 to 1 to grow up. So firstly we need to integrate this Inner part with respect to X. And then lastly who integrate this outer part with respect to why six? So integrating the inner part you're from seem to wanna. And then the way I'm gonna radio functions are going to separate it. E to the X, the E to the negative Y. So this is the same as what we're given there. So it's just to make the integration simpler. So the way we integrate this is we treat our Y as a consent. The reason being we are integrating with respect to X. So everything which is not X. We treat as a constant. So E to the problem is one is a constant so you E 24 minus Y is a constant. So or treat constants. When we're integrating you know, we can figure out the constant and then was left inside these E to the power X. And then we are integrating with respect to the mix. Okay, now we write our constant as it is. If we integrate into the vortex which remains either by X. You can my family's by differentiating these little projects to see that if you used to hear it to the Vox No, After we integrate we need to apply our boundaries from 0-1. So again, if we, if we play our boundaries, we're gonna have into the problem which is just eat minus, she played in Zero, we're going to have eaten Porzio which is one. So you have -1. So this becomes our inner integral. Now we need to find our authentic girl from 0-1 with respect to why? So come here we see from 0 to 1 we are now integrating each of the loneliness. Why? And do we have A -1 is a constant Again, we'll figure that out human as one. Yeah. Was left inside for us to integrate. Yes. Okay, progressing further with this. If we integrate into ominous why we'll see that we have minus E. That's why then we want to play our boundaries. Yeah, From 0 to 1. Okay. Weird question again. And then here, so when I first Plugging one as an upper bound so yeah minus e minus one and then we have minus minus E to form a zero which is just sitting there and then here we have okay eight minutes one. And here we have surfaced lee looking at this year, we have many E to the power in a zero. So everything to the poor zero. It's once we're gonna afghanis one minus and the minus become positive one. So you're going to have one. My maintenance mm to all this one. Yes. So this will become more answered Thank you

And this will do will show how to compute the inte grams. So you have an interest in integral. Which we can also call dump integral. Give our function again. Also what A Which is the idea X. So firstly when we integrate this we need to find the integral of this inner part. And then lastly we find the broke this outer part. So firstly we integrate the twi and then lastly integrate with respect to X. And here in when you see that high preaching, this general familiar. So this will come in handy doing our calculation. Okay so starting in its inner part we have boundaries. You have E to the power next. What you with respect to Why? So you have to know that we're integrating with respect to Y. So everything in our function which is not why we treat it as a constant. So meaning this whole B to the X squared is a constant. So we know that when we integrate have any consent and I have this then we need to apply our boundaries so from zero to thanks. Okay now plugging in to see that we have into thanks to the a squared fly by next we're planning zero. Everything is gonna be zero. So I won't write zero there. So this becomes our inner integral. Now we need to find out hunting girl from 0-1 with respect to X. Come Here. I Promise You one. Yeah it's into the power X squared. So this what you wrote here is because our inner integral what we got there and this is now with respect to X. So now here you see that that's where we're going to use whatever green there sweet year our you X. Is X squared. And now we need to find the first day video of us. So if I can right here, come here I'll see our us because it was too X squared and our first derivative Will be two X. But if you take a look here we have X. We're going to have uh two X. So we need to transform this X. So that it becomes two X. How do we do that here? So Yes, the same one you want to X according to our general formula. So if we put half outside the year so they're half and to counsel which we have here. So now that we have this function which is similar to what I've got here because These two X We have this first integral. So it's our first derivative of extra too. So now we can easily write the integral. All these is E. X. to the power two and 3 Play the bodies from 0 to while. Okay, plug in tv boundaries my hands. E to the poor one squared. She's meant this. He could for one clicking sale minus bad. So this becomes your no concept. Thank you for your time

Hello in this video will show how to compute these into girls. So we have our function here. She's a function of X and Y. And you're given our boundaries. So from 01 and two. And again we're given a D. H. D. Y. So firstly what we need to do is we need to integrate this inner part with respect to X. And then afterwards will integrate this outer part with respect to why? So starting with the inner pod, you see that here we have The internal from 0 to 2 and they don't give up function. Which is this function here X. Y. The X. Okay. So progressing further with this here, we see that we're integrating with respect to X. So our why we treat it as a constant. So everything which is not an X. Inside of function will be treated as a constant. So treating why is a constant and uh the integral of eat with poor X. We know that it remains into the X. So you're gonna have our constant and you have to eat the pork. It's as it is the integral of texas spread of the two. And then lastly why which is a constant. We're going to have White house and then we need to apply our Founders sort of from 0 to 2. Okay applying to we're gonna have y. So remember we were integrating with respect to X. So our boundaries are applying to X. So don't be confused by planning to Y. Just remember you're integrating with respect to what and then you need to plug in your boundaries on that variable. So replacing them, we're going to have it with a part two, find this, You will have 2-4. 2/2 minus why? Yeah, minus We applied the lower bound. Gonna have why? Each of the posy oh and then if we put plugging zero on the west gonna have zero. So I won't write zeros there. Just leave it like this. Okay, now we need to simplify further. All right, this is it is you could simplify E to the power to I'll just leave it as it is now -2 to the part which is Ford very right to get to. And then here we have managed to Y and then here we have to deposit which is one. So we're gonna have minus why? Here we simplify further. If this it's too You're gonna have -3. Y. So this is our in integral. Now we need to find out integral from 0-1 with respect to why? So, coming you say from 0-1, what's all function? Now It's not this one and this too, that's three. Y. And now it is with respect to why integrating this? So our E to the party is a constant. We can write it with two, integrating Why do we have what is it about 2/2 minus two wide -3 Y. to the statue. Uh huh. Two and then again we are playing our boundaries 0 to 1 as we adhere. Okay, no plugging in our boundaries. Again we start with one, you're going to have it to the cartoon don't Which represents an application yet minus ea for plugging one. When I have minus two in the a for plugging one, we're going to have 300 Part two. So the one will still remain onto the porch was too many as one. Then we have three of the two. Yeah. Again uh Zero. Everything will become zero there. So instead of writing manners zero I'll just leave it like that because There's no point of us writing zero. Could write to feel one too. But yeah, I'll just leave it like this. So what will be left of us here? It's just maybe to simplify further. So here we're gonna have um half Be into the part two. And here we have uh minus two minus 312. You get -7 or two so you're minutes seven cheap. So this becomes your final answer. Thank you. Feel tired

In this video who show how to integrate he stopped integral here here or can also create an eater integral. So here are function is a function of X and Y. And then we give in the order with which is txt white and then we also given our boundaries there. So the way we approach this is firstly we integrate this in a part with respect to thanks. So be making sure here that Integrate with two x. And then lastly you integrate out apart with respect to why? So starting with the first one we have himself to three and then we have into the power X plus one. So what I'll do is how separate these. So this is the same is what we are given there. So this is just too make the intervention little bit simpler. Okay so now the way we integrate this is what's important is to know that we fdx which is with respect to X. So everything in our function which is not hits which did as a constant meaning it with the bar Y is a constant. So we know that constant. We can factor them outside of integral. So All right it is this and we'll be left with into the power X. Okay and then the ctx so continue further with this integration. They have a trip boy is not constant and then into the politics. The integral will remain into the politics. Now we need to play boundaries from 0-3. Playing with the boundaries. Starting within a bound. We have e to the par three -E to the cause um which is one. So this is our inner integral. Now we need to find out the integral from 0 to 2 with respect to why. So we come here We interviewed him from 0 to 2 and now we are integrating the function which we found there it is things through the district to why. Remember we can affect our constant and then we'll be left. It's into the call why integrating this right? Our consent Into the 4 3 males, one integrated into the ball. I was studio into the problem why We can play your boundaries from 0-2. Okay. Yeah. Again our constant. If we apply to when I have E to the power to If we apply you have -1. Yeah. So okay you can just remove that, hold the skate bracket, we make it this one and then you could simplify this. We could just leave it like this and this will become your final answer. Thank you for your time.


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