In this video I go over the substitution rule for integrals and show why it is necessary for solving integrals such as integral of 2x·sqrt(1+x2). I also go over the proof for the substitution rule in which I show that it involves permissible substitute variables and differentials directly into an integral.
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The Substitution Rule for Integrals
Our Antidifferentiation formulas don’t tell us how to evaluate integrals such as this:
To find this integral we use the strategy of “introducing something extra” by changing from the variable x to a new variable u.
The Substitution Rule
If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then:
Proof:
Thus, the Substitution Rule says that it is permissible to operate with dx and du after integral signs as if they were differentials.