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Integration by substitution

  1. Mar 12, 2017 #1
    1. The problem statement, all variables and given/known data
    Suggest an integral that is reduced to a rational function integral when this substitution is used:
    ##a)## ##t=\sin x##
    ##b)## ##t=\sqrt[6] {x+5}##
    ##c## ##\sqrt{1-9x^2}=-1+xt##
    2. Relevant equations
    3. The attempt at a solution

    I found this to be a very interesting problem and wanted to check my results with you. For the first part i think that
    a) ##\int \sin x\cos x \, dx## is a good idea couse when we introduce the given substitution we are left with ##\int t \, dt## which is a rational function, right?
    For the second part i have some struggles but i think that
    b) ##\int \frac{\sqrt[6] {(x+5)^5}}{6\sqrt[6] {x+5}} \, dx## would reduce to ##\int t \, dt##
    As for the part c) that i the second Euler substitution and the integral that is suited for it should be any integral of the form c) ##\int \sqrt {ax^2+bx+1} \, dx##
    How does this seem to you? Any comments? Any feedback is appreciated
     
  2. jcsd
  3. Mar 13, 2017 #2

    Orodruin

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    The easiest way is to start with an integral of a rational function and do the substitution in the other direction ...
     
  4. Mar 13, 2017 #3
    I'm gonna try that as well, could you take a look at what i came up, to make sure i got it right or wrong?
     
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