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How do I know what to substitute(u-substitution)

  1. Jul 31, 2014 #1

    MMM

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    1. The problem statement, all variables and given/known data
    Hello, I'm having trouble determining what I make u equal with most integrals that revolve around the inverse trig functions.

    I knew what to let u equal in this problem ##x/\sqrt{1-x^4}## I let it equal ##x^2## and correctly solved the problem.

    The problems I'm having trouble with, are problems like ##1/(a^2 + x^2)## the book said to substitute u = x/a and when I did that I found the correct answer. Sadly I had no idea that is what I was supposed to substitute.

    Here is another problem I had no idea what to substitute ##1/(a^2 + (b^2)(x^2))## the book said to substitute u = bx/a and when I did that I got the correct answer. I just need some guidance to understand the kinda substitutions I need to make regarding integration involving inverse trig functions. Any help is greatly needed and highly appreciated, thanks.


    2. Relevant equations



    3. The attempt at a solution
    I solved the stated problems with the book's help on what to substitute. I just had no idea on to what to substitute.
     
  2. jcsd
  3. Jul 31, 2014 #2
    You are looking for something you know how to integrate. In this case when you see something like ## 1/(a^2 + b^2x^2) ##, your first instinct should be that it looks kind of like ## 1/(1 + x^2)## and so we want to substitute to put it on that form.

    Here we note that
    \begin{equation*}
    \frac{1}{a^2 + b^2 x^2} = \frac{1}{a^2(1 + \frac{b^2}{a^2}x^2)}= \frac{1}{a^2} \frac{1}{1 + (\frac{b}{a}x)^2}
    \end{equation*}
    and the substitution suggests itself.

    You see these things easier and faster as you get more and more experience.
     
  4. Jul 31, 2014 #3

    MMM

    User Avatar

    Thanks for the help. I understand it now!
     
  5. Jul 31, 2014 #4
    To practice and since you knew how to integrate ##x/\sqrt{1 - x^4}##, how would you tackle
    \begin{equation*}
    \frac{ax}{\sqrt{b - cx^4}}?
    \end{equation*}
     
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