a lecture on integration by substitution
The Method of Substitution (or Change of Variables)
How to use it: In this method, we go from integrating with respect to to integrating with respect to a new variable, , which makes the integral much easier.
Find inside the integral the composition of two functions and set “the inner function”.
We also write .
Substitute everything in the integral that depends on in terms of .
Integrate with respect to .
Once we have the result of integration in terms of (), substitute back in terms of .
The method is best explained through examples:
We want to find . The integrand is , which is a composition of two functions. The inner function is so we set:
Substitute into the integral:
The following are typical examples where we use the subsitution method:
The inner function is and . Thus . Substitute:
The inner function is and . Therefore:
Inner and . Thus:
Now another integral which is a little more difficult:
The inner function here is and .
This function is also a typical example of integration with substitution. Whenever there is a fraction, and the numerator looks like the derivative of the denominator, we set to be the denominator:
As in the example above, we set , :
Here the inner function is and . Thus
Some other examples (solve them!):
|Title||a lecture on integration by substitution|
|Date of creation||2013-03-22 15:38:29|
|Last modified on||2013-03-22 15:38:29|
|Last modified by||alozano (2414)|