Current UK exam textbooks pass over proofs and mathematical discussions in a hurry to show the ‘how to’ of exam questions.
Integration by substitution is a little more than just backwards chain-rule and deserves a fuller treatment.
Suppose that there exists a function g, of another variable , such that and let, . So that,
Now, by the chain rule,
Formula and completing the square – basic stuff for School maths students expressed loud and clear in Carr’s Synopsis of Elementary Results in 1880 – find a proof of 45 by using ideas in 47 and teach yourself to be a mathematical genius.