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U Substitution

Here we are trying to find a little function u(x) which fits inside the bigger function f(x) something like this:

Formula

where we have defined

This can make large integrals “telescope” down to very simple ones. The hard part is learning to recognize the little function u(x), since it likes to hide (and math professors think this is real cute and funny, so watch out!). To deal with this (usual advice): consider lots of examples!

Examples
|| 1 || 2 || 3 || 4 || 5 || 6 ||

Example 1

Set

then

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Example 2

Set

which makes the integral (see Integration by Parts or Integral Table)

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Set

then the integral becomes

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This one is trickier, and requires some skill at recognition. There don't seem to be any clean derivatives of the “inside”, but that must be u(x) since it is raised to such an ugly power, so let's try

This is real nice, but what about the x cubed term? Well,

and that leaves just the limits:

now our integral packs up nicely:

Set

then

Example 6

Set

Then

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Recommended Books

The classic calculus problem book - very light on theory, plenty of problems with full solutions, more problems with answers

A simplified and updated version of the classic Schaum's Outline. Not as complete as the previous book, but enough for most students

u substitution, substitution, integrals, technique