Theory

Theory 1

The method of -substitution is applicable when the integrand is a product, with one factor a composite whose inner function’s derivative is the other factor.

Substitution

Suppose the integral has this format, for some functions and :

Then the rule says we may convert the integral into terms of considered as a variable, like this:

The technique of -substitution comes from the chain rule for derivatives:

Here we let . Thus for some .

Now, if we integrate both sides of this equation, we find:

And of course .

Extra - Full explanation of -substitution

(1) Chain rule for derivatives:

Let be a function and its derivative. Let be another function.

Using primes:

Using differentials:

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(2) Integrate both sides:

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(3) Introduce ‘variable’ from the -format of the integral:

Treating as a variable, the definition of gives:

Set the ‘variable’ to the ‘function’ output:

Combine these:

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(4) Substitute for :

This is “-substitution” in final form.