Crystal Clear Maths's video: Jim Coroneos 100 Integrals 013 2x-1 x 2x 3 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 thirteenth problem is to evaluate ∫(2x-1)/(x²+2x+3).dx Because the numerator has degree one less than the denominator, we expect a logarithmic function to be part of the solution. Because the denominator cannot be factorised (within the real number system), we cannot use partial fractions. Therefore, we rearrange the numerator to create two separate expressions/integrals. The numerator of the first one will be the derivative of the denominator ... and the second expression will contain the 'remainder.' The first integral produces the logarithm that we expected. The second integral, after completing a square, produces an inverse tangent function. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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.