Crystal Clear Maths's video: Jim Coroneos 100 Integrals 016 1 x 1-x 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 sixteenth problem is to evaluate ∫1/[x²√(1-x²)].dx This expression does not fit the 'normal' patterns that we would look for ... chain rule with f'(x).f(x)^n, or an inverse sine pattern (which would require just √(1-x²) in the denominator). We therefore simplify the radical by using a trigonomtric substitution. We would like a perfect square within the radical, so the identity that is of use to us is either sin²x = 1 - cos²x or cos²x = 1 - sin²x We choose to substitute x = sinθ. After simplification, the resulting integral is ∫cosec²θ.dθ ... which would be written ∫csc²θ.dθ in the USA. This is a lesser known integral in schools here in NSW, Australia. Not only is ∫sec²θ.dθ = tanθ + C, but ∫cosec²θ.dθ = -cotθ + C You can check this by finding the derivative of -cotθ. Substituting x = sinθ in this result provides our answer in terms of x. This little integral took us on an interesting journey where we learned about trigonometric substitution and about ∫cosec²θ.dθ. It is worth knowing both. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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.