5.5 The Substitution Rule. The Substitution Rule. If u=g(x) is differentiable function whose range is an interval I , and f is continuous on I , then The Substitution rule is proved using the Chain Rule for differentiation.
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The variable of integration must match the variable in the expression.
Don’t forget to substitute the value for u back into the problem!
We computed du by straightforward differentiation of the expression for u.Substitution rule examples
The substitution u = 2x was suggested by the function to be integrated. The main problem in integrating by substitution is to find the right substitution which simplifies the integral so that it can be computed by the table of basic integrals.
This rewriting allows us to finish the computation using basic formulae.Example about choosing the substitution
Next substitute back to the original variable.
If g’ is continuous on [a,b], and fis continuous on the range of u=g(x) then
Find new limits
Don’t forget to use the new limits.
aIntegrals of Even and Odd Functions
An odd function is symmetric with respect to the origin. The definite integral from -a to a, in the case of the function shown in this picture, is the area of the blue domain minus the area of the red domain. By symmetry these areas are equal, hence the integral is 0.