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Integration by Substitution
Many expressions may be simplified by means of a suitable substitution i.e. change of variable to recognisable forms ie. one of the standard forms.
The following example illustrates the steps involved in the substitution method.
Explanation of Steps:
Step 1: Using the substitution provided, differentiate with respect to x.
Step 2: Replace terms in the integrand involving x with terms involving new variable,u. The operator dx has to be changed to du accordingly. Simplify if necessary.
Step 3: Integrate the integrand with respect to u.
Note that the new integrand should be easier to integrate (if not, it defeats the purpose of carrying out the substitution). Otherwise, check steps 1 and/or 2 for mistakes.
Step 4: Replace all the terms in u with that in x.
Note: Under the syllabus, the substitution to be used will be given to you in the question if you are expected to integrate using a substitution.
However, even if the substitution is not given, you may still use the substitution method if you know the appropriate substitution to use. (But there will usually be another method of doing it.)
Example
By using the substitution u = ln x , find
∫[(ln x )^6 + 1]/x dx
Step 1:
Let u = ln x
du/dx = 1/x
dx/x = du
Step 2: ∫[(ln x )^6 + 1]/x dx
= ∫ (u^6 + 1)du
Step 3: = u^7/7 + u + c
Step 4: = (ln x)^7/7 = ln x +c
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