Crystal Clear Maths's video: Jim Coroneos 100 Integrals 019 1 x x -a 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 http://crystalclearmaths.com/wp-content/uploads/Jim-Coroneos-Obituary.pdf. 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 http://bbujeya.blogspot.com.au/2014/03/100-integrals-from-coroneos.html. 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 nineteenth problem is to evaluate ∫1/[x√(x²-a²)].dx This integral only differs from the previous integral only by the reversal of the difference between squares within the radical. In other words, the expression inside the radical is x²-a² instead of a²-x². We simplify the radical by choosing an appropriate trigonometric substitution. Because of the "-" sign within the radical, I chose to use the Pythagorean Identity a²tan²θ = a²sec²θ - a², and therefore substituted x = a.secθ. [You could, equally well, have chosen the identity a²cot²θ = a²cosec²θ - a², and substituted x = a.cosecθ ... or x = a.cscθ in the USA!]. PLEASE NOTE that I glossed over a potential problem here! The square root of tan²θ is actually the absolute value of tanθ because the radical takes the positive value only. This means that you should be very careful of the domain in which you are integrating. If this was a definite integral, the limits of the integral in x, when converted by the substitution x = a.secθ ... or, rather, θ = cos‾¹(a/x), would provide the limits in θ. You should then note how tanθ behaves in this θ domain. Since this is an indefinite integral, however, I should have discussed BOTH possibilities and noted the appropriate domains for each ... but I felt that the video was going to be too long anyway. I will leave this as an exercise for you to deal with. If you contact me privately, I would be happy to explain further or, perhaps, to make another (explanatory) video to append to this one! After simplification, we are astonished! The resulting integral is a surprisingly elementary (1/a)∫dθ. It is integrated easily and then we substitute for θ to produce our answer in terms of x. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For more information about mathematics or how to study, visit my website, Crystal Clear Mathematics at http://www.crystalclearmaths.com/ If you wish to be kept up to date with what I am producing on the website (ad free, spam free, cost free mathematics and study materials), please add your name to the mailing list there. Download my FREE 32 page PDF "How to Study" booklet at http://crystalclearmaths.com/wp-content/uploads/2015/12/How-to-Study-Mathematics-V3.pdf Best wishes for your study and your mathematics! Thank you.