Crystal Clear Maths's video: Jim Coroneos 100 Integrals 030 1 3 5cosx 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 thirtieth problem is to evaluate ∫1/(3 + 5cosx).dx As with the previous video, when we integrate a function that is in the form of a fraction containing a simple trigonometric ratio (and sometimes more than one), the best substitution to use is usually what we call "t formulae" or "half angle formulae." You will notice, however, that this integral is different from the previous one in that the 3 and 5 have swapped positions. This means that, after substitution, the integral has a DIFFERENCE between squares in the denominator rather than the SUM of two squares. In the previous video, the sum of squares formed a pattern that gave rise to an inverse tangent function. In this video, the difference between squares is resolved using partial fractions. The two resulting integrals produce logarithmic functions that can be combined to form a rather interesting solution. In a later video series I will be deriving and explaining how to use these t-formulae in considerably more detail. In this video I simply use them to evaluate the integral. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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.