Integration: By Substitution

[ method ][ example#1 ][ example#2 ][ example#3 ]





The Substitution method(or 'changing the variable')

This is best explained with an example:

substitution problem#1

Like the Chain Rule simply make one part of the function equal to a variable eg u,v, t etc.

substitution variable

Differentiate the equation with respect to the chosen variable.

substitution derivative

Rearrange the substitution equation to make 'dx' the subject.

substitution rearrange

Substitute for 'dx' into the original expression.

substitution dx

Substitute the chosen variable into the original function.

substitute chosen variable

Integrate with respect to the chosen variable.

integrate with respect to t

Restate the original expression and substitute for t.

final equation

NB Don't forget to add the Constant of Integration(C) at the end. Remember this is an indefinite integral.


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Example #1

substitution problem#1


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Example #2

substitution problem#2


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Example #3

substitution problem#3




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