Crystal Clear Maths's video: t-Formulae 03 Used to Evaluate Certain Integrals

One of my subscribers asked me to explain what t-formulae are, and how they are used in trigonometry. This is the last of three videos that I created in response to that request (thank you, Kristyn). Because t-formulae allow us to convert trigonometric functions into fractions involving polynomials, we can use them to change certain trigonometric integrals to 'integrals of fractions containing polynomials.' This can often (but not always) be an advantage. Such substitutions are particularly valuable when the original integral only involves single instances of trigonometric functions and no powers. In fact, this is when t-formulae are almost exclusively used. In other words, we look for integrals that involve uses of sin(x), cos(x), and tan(x), but not powers of these functions. The fraction that results will contain two polynomials. These may produce a logarithmic structure, or may require a division to take place, or may lead to the use of partial fractions, or may even produce inverse tangent functions, for example ... but many of these structures are almost instantly recognisable and experienced mathematicians know how to 'handle' them. In this video, I provide a 'typical' example ... ∫1/(3 + 5cosx).dx ... which is Jim Coroneos' 30th integral (see my YouTube playlist, https://www.youtube.com/playlist?list=PLJ0RnXkLrsTRPyurk9ZsAQM0atYAKjsPB). Because the complete solution of this integral has been accomplished in another video, I only spend sufficient time with the integral to show how the use of t-formulae allows us to convert it to a much more useable and recognisable form. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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!