Jim Coroneos' 100 Integrals ~ 028 ~ ∫tan³x.dx

Jim Coroneos was a remarkable teacher and a wonderful human being. I had the great privilege of studying from his text books in my senior high school years and, during my first years of teaching, working alongside him as a colleague for a while. Sadly, he passed away almost exactly 10 years ago, in 2005, but you may read something of his life at crystalclearmat....
In one of his more advanced texts, he provided a list of 100 Integrals to challenge his students. This list is now used by mathematics teachers and students world wide. The complete list has been produced on a few websites. You may like to obtain a copy from bbujeya.blogspo....
Partly to honour Jim, and partly to fulfil an international need, I have decided to produce 100 videos, showing how to solve his 100 integration 'problems.' I hope you find the videos useful!
This twenty-eighth problem is to evaluate ∫tan³x.dx
This is a surprisingly easy and very elegant little integral. This is because tanx has a wonderful property in that it is linked to sec²x in two different ways. First, you will remember the trigonometric identity 1 + tan²x = sec²x (which produces tan²x = sec²x - 1). Second, the derivative of tanx is sec²x. We are able to use both of these facts in evaluating this integral!
I will leave it to you to watch the video to see how it is done. There is even a lovely little twist where we find that ∫tanx.dx is a logarithmic function! I hope you enjoy this delightful little problem.
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Best wishes for your study and your mathematics!
Thank you.

[@MrVoayer]

Not only did we learn integration technique, we learned (or were reminded of) some nice properties of tan function, too. That was delightful. As well as were two integrals in the end, each with its own method of integration. A few important and valuable tricks to be learned from this "delightful little integral". Thank you for showing us the workout !

[@CrystalClearMaths]

+MrVoayer Again it is lovely to hear from you, friend. I am glad that you enjoyed this video. The integral is, indeed, quite a delightful one ... containing some lovely surprises and insights for us. Warm regards, Graeme

[@aram5642]

Sheer wizardry! Pleasure to watch and to get perplexed as you pick up all the identities out of the black

[@CrystalClearMaths]

Thank you, friend.