Jim Coroneos' 100 Integrals ~ 005 ~ ∫sinx.sec³x.dx (Three Methods)

[@MrVoayer]

Valuable explanation and enlightening teaching of how to think and how to connect integral and derivative. It requires some practice to become versed in various methods but it is important to stay open minded and be prepared to accept other possibilities. "u substitution" is probably the most widely used "by the book" method of solving this particular integral. But two other methods are more than valid.Thank you, Greame, for broadening our knowledge.

[@CrystalClearMaths]

+MrVoayer You are such an encourager, MrVoayer! You know my background and struggles and take the time to respond warmly to each video. Thank you.

[@jazcreations]

Hi I was just wondering if you can put the same integral in the form: [tan(x)xsec^2(x)]dx and integrate from there. I tried doing this and got a different answer. I don't know if I did it wrong or if there is a reason you cannot do it this way. Thanks!

[@CrystalClearMaths]

Hi JC, and thank you for your query. Your method is actually "better" than the one I use here (and gives the same result). There are often a variety of ways in which these integrals may be evaluated. I have usually been using just one method ... and not always the "best" one. The reason is that, if I can see a principle worth explaining to students (or if there is a method that most students are likely to adopt), I will adopt/explain that method. I am realising that that has not been a good policy. I need to create multiple videos, or longer videos, to illustrate alternative methods. In this case, your solution would be to take ∫sinx.sec³x.dx and, realising that secx = 1/cosx and that sinx/cosx = tanx, convert the integral to ∫tanx.sec²x.dx. This can be written as ∫tanx.d(tanx) = (tan²x)/2 + C This solution is not at all different from mine ... which is 1/(2cos²x) + Cₒ. Note that I have used a different constant from yours. This is because 1/(2cos²x) + Cₒ = (sec²x)/2 + Cₒ = (tan²x + 1)/2 + Cₒ = (tan²x)/2 + 1/2 + Cₒ = (tan²x)/2 + C as long as C = Cₒ + 1/2. So, you see, the difference is only in the constants. The definite integrals would give the same results. You may like to try that. I very much appreciate your perceptive comment and the fact that you took the time to contact me. I have not produced videos for a while due to some series illnesses in the family but, when I resume, I will create some more of these integration videos, including one illustrating this alternative method that you have identified. Warm regards and best wishes for your studies! Graeme PS I just had a look at your intro video. You produce some seriously clever and creative work, friend! Very impressive! I wish you great success with it.

[@jazcreations]

Thanks so much for your fast and in-depth reply! Your methodology behind the constants makes complete sense, so thankyou for the clarifiction. The intro video that you saw is actually from my dad's channel as I am logged in on his account and decided to leave a message without changing. I will pass your compliments on to him! My name is Jess and I have been studying for my trial exams that are coming up soon and my teacher recomended your videos. They are very helpful! I am sorry to hear about your troubles and I hope your family is okay. Again, thankyou for your response, I am very grateful and honoured to hear you're planing to make a video using this method! Kind Regards, Jessica

[@CrystalClearMaths]

Thank you very much, Jess ... and all the best for your trial exams. I am quite touched that your teacher has recommended my channel. Do you study in NSW, Australia? You will find that different integration methods will often produce seemingly different results, but the differences are in the constants! This is an area worth exploring, so I encourage you to think up different ways of evaluating some of these more difficult integrals. You will develop much stronger manipulative skills if you do so. I also appreciate your concern and kind thoughts for my family. Thank you. I won't go into details here, but some of my family have a difficult journey ahead. Kind regards and best wishes for your studies, Graeme

[@jazcreations]

Yes, I attend hunter school of the performing arts in Newcastle. When watching your video's I often look at the question and try it myself. From the ones i have done, I often use a different method than what you use and sometimes get to a different solution. I will have to go back now and see if I can manipulate what I get into the same result! Thank you for the advice and the hope for my exams, I hope I do well! Cheers, Jess

[@CrystalClearMaths]

I have relatives who taught (and teach) in the Hunter region. Currently, I live in the Shoalhaven, south of Sydney. Please pass on my regards and thanks to your teacher. I hope you enjoy reviewing some of your past solutions. I am sure that you will learn a lot of good things in doing so! Best wishes to you, Jess. Graeme