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Math Help - U-Substitution Rules

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    U-Substitution Rules

    Could someone help me out by listing the order for U-Substitution Rules? I mean what over-rides what? I know that typically we

    1) Let u= inside of parenthesis
    2) Let u= exponents on e or a
    3) Let u= angle on trig functions
    4) Let u= denominator if no parenthesis

    Here is where the confusion comes in. What if you have fraction with a trig or inverse trig function in it? Which one should be "U"? What if you have a trig function and a square root in an equation? There has to be some since of organization in this. I know that I have only been working with substitution for less than 24 hours, but I am obsessed at picking it up...

    I spent 45 minutes on this one problem going back and fourth and finally I got the correct answer.

    I initially thought of substituting 1+x^2 but there was no way I could get the problem worked out. If you do this and you end up getting (1/2)du=x*dx I don't think it is legal to insert the (1/2) inside of the trig function?

    Thank You

    -qbkr21
    Attached Thumbnails Attached Thumbnails U-Substitution Rules-integrate-4.gif  
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  2. #2
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    Quote Originally Posted by qbkr21 View Post
    Could someone help me out by listing the order for U-Substitution Rules? I mean what over-rides what? I know that typically we

    1) Let u= inside of parenthesis
    2) Let u= exponents on e or a
    3) Let u= angle on trig functions
    4) Let u= denominator if no parenthesis

    Here is where the confusion comes in. What if you have fraction with a trig or inverse trig function in it? Which one should be "U"? What if you have a trig function and a square root in an equation? There has to be some since of organization in this. I know that I have only been working with substitution for less than 24 hours, but I am obsessed at picking it up...

    I spent 45 minutes on this one problem going back and fourth and finally I got the correct answer.

    I initially thought of substituting 1+x^2 but there was no way I could get the problem worked out. If you do this and you end up getting (1/2)du=x*dx I don't think it is legal to insert the (1/2) inside of the trig function?

    Thank You

    -qbkr21
    Your "answer" is that you are right and you are wrong. The list of substitution ideas is good, there's nothing wrong with that. But to find good substitutions can be something of an art form. As your example indicates, you need to look at the rest of the integrand to see what your substitution does to it.

    At the end of the day the rule is that there are no rules.: you use whatever works. How does that help you solve new problems? Well, you try what has worked before and if nothing does, then you need to play with different substitutions to see what works.

    That isn't much of a helpful answer, but you can always post problems (as you have been doing) and someone will give you an idea to try. The best way to learn how to find effective substitutions is to work a lot of problems and gain experience in what works when and why and what doesn't.

    -Dan
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