Problem 77 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{xe^x}{\sqrt{1 + e^x}}\ dx $

Answer

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Video Transcript

Let's try use up here. Let's take you to be swear, root one plus one ex. Hey, then do you using the chain rule here from capitalists? So here will get a tube down there then and then let's try to go ahead and rewrite thiss and actually, this's fine. Let's go ahead and and then a girl if we just put d'You hear. And then, of course, we'LL need to multiply by this to to give her that through there now by writing two times to you we already get here the ex DX and the square. So that accounts for all of this just by ready. So do you book? We see that there's this extra X factor egging out here, so we'LL have to go ahead and solve this rex so you square equals one plus Eat it, x. So subtract Juan and take that natural log on both sides. In that case, on the right, When you take the natural log, it cancels out with either the axles of inverse functions. So that's what we'LL go here that is our ex And then now for this new intern girl here that we have Let's go ahead and try in English. My purse So parts here. Let's take him. Use the letter W Here, Ellen, you squared by this one t w using the general. And then here Devi will just be to you. The is you say, using the formula for integration My purse Here let me go ahead on the next page to write this out. So we had to So you see in a girl the view and then we'LL go ahead and plug in our w would be So this is all coming from our immigration by parts. Yeah, And then now here we can go ahead and just distributed to And then here let me go ahead and pull of this four in the front of the integral And then the next step here would be to do long division or polynomial division. So go ahead and do the division here. And after doing that and the partial fraction decomposition we'LL have some really hear the Bruneian integrating it. The steps Here are two new polynomial division and then it's also faster that denominator as you plus one You minus one and then do partial fractions The composition. So after doing that you should get one one half over. You might this one and then minus one half over you plus one. So let's go ahead and integrate these on the next page and then now were integrating. So go ahead, distribute before and the last to the formal people supplied by the one half. And then So here were my so, too. That should not be a square. Some getting sloppy here at the end, You minus one and then plus to Ellen. And then you plus one plus he And then we use the definition of you from the first page. So just go ahead and simple file these so to you there. That's one plus either the ex and then here. Watch out for this for the second term. Right there is Ellen. You squared minus one. That's just ln off one plus either X minus one. Ellen. Either the ex in ashes X. So we put X here outside the radical and then minus for you their selections. Four times radical Cool. And then we have plus or sorry here. And it's still a minus minus now. Alistair here to natural live absolute value, you could drop the absolute value for this one. But if you're not sure just going to leave it in there and in the very last one, you can also track the absolute value here for the same reason. But this one gets a plus one instead of the minus one that we saw out there and we add that constant of integration sees, and that's your final answer.