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Trigonometric Substitution for Integrals: One to One Function
In my earlier videos I went over trig substitution for integrals as well as some examples but I did not explain in too much detail why, when defining the trig substitution, the domain needs to be restricted to enforce a one-to-one relationship. In this video I go through extensive detail in illustrating the reason for restricting the domain as well as the different types of trig substitution cases.
Basically since trigonometric functions are cyclical, when defining the variable x as a trig function the problem arises in which there are multiple values of the trig angle that bring up the same x value and thus the integral will have multiple values (that are still identical) for the same x value. This is a problem since most concepts in mathematics, including integrals, are defined only for one-to-one functions and thus we have to restrict the domain of the trig substitution while still ensuring that the domain of the x variable is still maintained. This is a pretty extensive example but it will make trig substitutions a lot more clear so make sure to watch!
Download the notes in my video: http://1drv.ms/1FIIBA1
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/trigonometric-substitution-for-integrals-one-to-one-function
Related Videos:
Trigonometric Substitution for Integrals: Example 4: http://youtu.be/v3tHdU63Vio
Trigonometric Substitution for Integrals: Example 3: http://youtu.be/Tfl90TqW3jA
The Area of an Ellipse (and Circle): http://youtu.be/7hSkKPQA71s
Trigonometric Substitution for Integrals: Example 1: http://youtu.be/lpp6YLWB2GM
Trigonometric Substitution for Integrals: http://youtu.be/2pWvGXwtVJo
Inverse Functions Part 2: One to One Functions and the Horizontal Line Test: http://youtu.be/iSNuplUWrbA .
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