Problem 84 Hard Difficulty

We know that $ F(x) = \displaystyle \int_0^x e^{e^t}\ dt $ is a continuous function by FTC1, though it is not an elementary function. The functions
$ \displaystyle \int \frac{e^x}{x}\ dx $ and $ \displaystyle \int \frac{1}{\ln x}\ dx $
are not elementary either, but they can be expressed in terms of $ F $. Evaluate the following integrals in terms of $ F $.
$ \displaystyle \int_1^2 \frac{e^x}{x}\ dx $ and $ \displaystyle \int_2^3 \frac{1}{\ln x}\ dx $

Answer

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Video Transcript

So let's go ahead and express these two definite rules in terms ofthe this capital f of X up here. So let's start off with the first Integral wanted to either the ex over X t x And for this one, let's take a use of so that do you equals one over X. The X and I could rewrite this inter girl is Ellen of one tellin of two. If I just write to you, I'm already getting this one over x t x so I just need e to the X But in terms of you. So I get that by taking this original use up. I've raised both sides by using e so exponentially ation No. And then I want eating eggs. I'd do it again. So we have e to the t to the U. And then we don't natural lot of one zero. So we have zero Ellen of two and then using this right here. This is the function, the same in and roll that's expressed up here, that he it doesn't matter whether it's it's here. You here because that's just a dummy variable. So this is just capital F of Ellen of two. So this is for the first answer the first, integral. And keep this answer and mind because we'll use this idea for the next part. So let's go to the next page. This was the second inaugural. So here, let's go ahead and do the same use up. And this time, let me I don't have a one over X over here, so I don't like this x. So let me go ahead and solve this for X. So I have either the U equals X and then I'll solve this equation here for DX. So we have inner girl Ellen to Ellen three and then one over X. So that's one over you and then d X Dash is either the you do you and this is the same. Not the whole integral. But if you just look at the end of Randy either the u over you, do you That was the same as the inner rule with yourselves. So we'll actually use the earlier work here to get the answer. So first, I would just use the property of the definite no girl to rewrite this because we would like the arrows on the bottom so we could use the earlier part of this problem. So using your properties of the log, basically, I'm just using this property here. And a girl from a to C. Is it a girl from A to B plus integral from BBC. And then you could just subtract one of the inner girls here. And this is where that minus is coming from. So for these last two hundred girls that are very similar to each other, you could either do the same exact substitution we did for the previous problems. So here you would do Ellen of X again. But after you do that, we basically saw what happens. You just take happen. Lf and then the natural log of whatever's up here. So our own case, it's natural log natural log of three. And then here we have f natural log natural Margot too. And that's final