I’ve always found integration by substitution satisfying, and the tangent half angle substitutions are particularly satisfying. These are also sometimes called “The Weierstrass Substitution” despite no evidence of it in his writings, and it have being used long before in the work of Euler.

It is guaranteed to transform a rational function of sine and cosine into an algebraic rational function that can be integrated (However, it should be noted that sometimes there will be neater, less laborious ways to evaluate a given integral).

I hadn’t spent too long on this during the course (it isn’t explicit in the AQA course), but contact with them has led me to believe that students could be guided through using these substitutions. Because of this I have produced a short guided work through sheet.

My students enjoyed doing these which is nice 🙂

File for your use is here.