Integration by substitution

In calculus, integration by substitution is a method of evaluating an antiderivative or a definite integral by applying a change of variables. It is the integral counterpart of the chain rule for differentiation. The solution for a system can be usually found by utilizing graphing, but it may not be the most precise method. For a definite integral, it can be shown as follows:

Where   so that .

 Steps 

 Let a variable equal part of the integrand, so that its derivative will cancel with the other part of the integrand
 Apply the substitution
 Evaluate the integral in terms of the new variable

 Definite integral example 
Consider the integral

Let  to obtain  and . In this case, the  variable is not present, so the 1/3 can be factored out of the integrand. Since the integral is now in terms of , the bounds of integration (1 and 3 in this case), must be plugged in to the substitution u=3x. So the new bounds of integration are 3 and 9 to obtain,

The antiderivative of  may also be found using integration by substitution and ends up being .

References

Category:Calculus