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) The substitution method comes from the chain rule for derivatives. The rule simply comes from integrating on both sides of the chain rule.

Setup: functions and .

Let and be any functions satisfying , so is an antiderivative of .

Let be another function and take for its independent variable, so we can write .

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(2) The chain rule for derivatives.

Using primes notation:

Using differentials in variables:

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(3)

Integrate both sides of chain rule.

Integrate with respect to :

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(4) Introduce ‘variable’ from the -format of the integral.

Treating as a variable, the definition of gives:

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

Combining these:

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(5) Substitute for in the integrated chain rule.

Reverse the equality and plug in:

This is “-substitution” in final form.