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Conisder the following problems $$\int e^x \Big(\frac{1-\sin x}{1-\cos x}\Big)dx.$$ or $$ \int e^x\Big(\frac{1+x\ln x}{x}\Big) dx$$

I have seen these problems (or simpler to them) in many various places but I do not have a specific reference for them. However, I did not see them in the standard calculus books like Stweart unless I missed it. The idea to solve these problems by using the following $$\int e^x (f(x)+f'(x)) dx=\int \frac{d}{dx} \Big(e^x f(x)\Big)dx.$$ Generally speaking, it depends about simplifying the function that are multiplying by $e^x$ and rewrites it as $f(x)+f'(x).$

For the seek of completeness, I will answer the first one by using the rule that I mention. Also, I was curious to see if there is another way to solve them by using one of the integral techniques.

00GB
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Here is my answer $$\int e^x \Big(\frac{1-\sin x}{1-\cos x}\Big)dx=\int e^x \Big(\frac{1-2 \sin (\frac{x}{2}) \cos (\frac{x}{2})}{2\sin^2(\frac{x}{2})}\Big)dx=\int e^x \Big(\frac{1}{2}\csc^2 (\frac{x}{2})-\cot (\frac{x}{2})\Big) dx, $$ This implies to $$\int e^x \Big(\frac{1-\sin x}{1-\cos x}\Big)dx=\int d \Big(- e^x \cot (\frac{x}{2})+C\Big)) dx= - e^x \cot (\frac{x}{2})+C.$$

00GB
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    I understand you may not be aware of this, but regarding self-answered questions (as yours seems to be) there is a lot of consternation on meta, particularly if the question . I can direct you to meta pages or have a conversation with you regarding why this is the case, but ultimately I have no issues with the answer, it's the question which needs some context and I can help you fill that in (apart from the attempt that the answer provides). – Sarvesh Ravichandran Iyer Jun 09 '21 at 23:17
  • @TeresaLisbon, Thank you. I really do not know about that. I thought there is no problem if ask a question and answered the question directly. I would love to have a conversation with you to avoid issue again in the future. – 00GB Jun 09 '21 at 23:42
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    I owe you one for your kind response, and thanks for editing your question. I'll give you some meta conversations regarding this issue soon, and if you wish we can discuss some problems with this issue. I would say this : the source of the question is important, so mention the source. You might be interested in why you want a second way of answering this question (something you don't like about the other method , for example?) Essentially, self-answered questions are held to higher standards because "sharing knowledge" means you should be in a great position to explain the... – Sarvesh Ravichandran Iyer Jun 10 '21 at 00:18
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    ... significance of such knowledge along with the motivation that made you ask it. You have to put on a questioner's hat for "why should anyone think about this question?" and the answerer's hat for : "what else can be useful when we think about this question?" and the answer itself. I quote : "For a self-answered question this point (of helping an OP, since the OP is the same as the answerer) is not really present and the question should be judged on its merits in contributing to the site as a repository of knowledge." I'll send you some meta links as I find them. – Sarvesh Ravichandran Iyer Jun 10 '21 at 00:21
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    Here are some meta-links : Posting a question to help others?,Posting PSQs with self-answers. I hope these are helpful. An extended discussion is also on the cards, but only when I'm free! – Sarvesh Ravichandran Iyer Jun 10 '21 at 00:34
  • @TeresaLisbon, Thank you very for the nice explanation. – 00GB Jun 10 '21 at 00:57
  • You are welcome! If you wish to improve your question along the lines I mentioned, you may continue to do so. Personally, I still feel you can add a source, and further motivation (may very well be personal, but is still important), and why you want a second method for solving the problem. Answering these should be enough to make this question compelling for a wide audience. – Sarvesh Ravichandran Iyer Jun 10 '21 at 00:58
  • @TeresaLisbon, Do you such an example? I have never seen a question with a lot of details as you mention – 00GB Jun 10 '21 at 01:16
  • This question on integrals was self-answered recently. Even it went through a lot of problems, but I think the way it looks now is good. For that question, the source is that they were thinking about a similar integral (which is mentioned), another method is required because integration-by-parts was failing (which has been explained in one line of computation). So both questions I asked you to address, were addressed in that question. – Sarvesh Ravichandran Iyer Jun 10 '21 at 01:18
  • The way I would put it is : for self-answered questions, the more the merrier. It's your task to make sure more people read the question, and the integral alone is not enough, so basically speaking we are just finding reasons as to why the question is interesting : only you will know the answer, that's why only you can edit the question to reflect these things. I personally don't think it's too much effort to add what these things, and since you are interested it should be a cakewalk for you. – Sarvesh Ravichandran Iyer Jun 10 '21 at 01:23
  • @TeresaLisbon, I added some explanations in my answer. Maybe, they are not enough. Thank you for your adivce. – 00GB Jun 10 '21 at 03:42
  • I told you it would be easy! You've just got a +20 from me, one each for answer and question. Thanks once again for helping me improve your question, and wish you a good day. – Sarvesh Ravichandran Iyer Jun 10 '21 at 03:45
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    @TeresaLisbon, you as well. – 00GB Jun 10 '21 at 03:50
  • @TeresaLisbon, I want to ask you a question about downvote if you do not mind. – 00GB Jun 14 '21 at 18:46
  • Yes, you can ask me (I was not available then, unfortunately, but I am now). – Sarvesh Ravichandran Iyer Jun 14 '21 at 22:36
  • @TeresaLisbon, No problem. Sometimes, I asked question and proived enough information but my question got downvote . I was woundering why this is happened. how can I know what is wrong in my question. Thank you – 00GB Jun 14 '21 at 22:44
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    Sometimes (and in my case, very often!) we get "unexplained" downvotes : these are just downvotes that come out of nowhere, and for no reason. For example, I get a downvote everyday on some random question or the other. If you can't explain the downvote, then usually it's either someone who is jealous/angry at something (need not even be you!) or could have even clicked the wrong button (I've done this before). In short, one downvote is nothing to worry about. When you get five of them in a short time, then you will know that something is wrong, but nothing for sure is wrong here. – Sarvesh Ravichandran Iyer Jun 14 '21 at 22:47
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    I'll give you an example : see this answer of mine and this answer of mine. I have no clue why they have downvotes, they were well written long explanatory answers. In my case, because I do some site-maintenance, I get revenge-downvoted quite often, to the point where I've stopped caring. – Sarvesh Ravichandran Iyer Jun 14 '21 at 22:49
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    If you want a more "specific" analysis, then you can link the question and answer you are referring to (if it's not this one, of course) and I can look at what's there and let you know what is going wrong. – Sarvesh Ravichandran Iyer Jun 14 '21 at 22:55
  • @TeresaLisbon, you have more information than I thought and you like to help those who do not know like me . This is really great idea. Thank you very much. Here is the link – 00GB Jun 14 '21 at 23:01
  • https://math.stackexchange.com/q/4172265/707884 – 00GB Jun 14 '21 at 23:01
  • My thoughts on that question (1) Maybe you can add links to the Bernstein facts you mention (the transfinite induction argument : it's best if there is a proof online) (2) You can add the definition of a perfect set. Apart from that, it seems that you have everything in place , so I can't say anything about the mathematics in the question, hopefully someone who knows set theory will comment and let you know. – Sarvesh Ravichandran Iyer Jun 14 '21 at 23:12
  • @TeresaLisbon, I will try to fix it. I have a request. How can I connect you in case if I have question related to some thing in this site. – 00GB Jun 14 '21 at 23:36
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    You have enough reputation : my chatroom is the Teresa Lisbon Chatroom. Drop a message there when you need me and I can be of help. One small thing, though : use @TeresaLisbon to call me so I get a message in my mailbox and I can reply back to you. I'm usually quite prompt. – Sarvesh Ravichandran Iyer Jun 14 '21 at 23:38
  • @TeresaLisbon, I will. Thank you so much – 00GB Jun 14 '21 at 23:57
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    You are welcome! – Sarvesh Ravichandran Iyer Jun 14 '21 at 23:57
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    @00GB (+2) See this meta answer. –  Jun 17 '21 at 03:39