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.

Try this,

Let,

then,

Suppose that there exists a function g, of another variable , such that and let, . So that,

Now, by the chain rule,

Hence,

i.e.

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