1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Indefinite Integration by exchange of variables

  1. Apr 26, 2010 #1
    1. The problem statement, all variables and given/known data

    This is only an example but I do not understand what they are doing...

    [tex] \int (4x+1)^{3} + (4x+1)^{2}+(4x+1) dx [/tex]

    2. Relevant equations

    [tex] \int f(u) du = F(g(x)) + C [/tex]

    3. The attempt at a solution


    [tex] u = 4x+1 [/tex]


    [tex] du = 4 dx [/tex]


    [tex] dx = \frac {1}{4}du [/tex]

    how did they get that dx was 1/4? There are no steps to explain this, it just lists them as having that value.

    So after substituting the problem should look like this:

    [tex] \int (u^{3} + u^{2} + u) * \frac {1}{4} du [/tex]

    which is this:

    [tex] \frac{1}{4}(\frac{u^{4}}{4} + \frac {u^{3}}{3} + \frac {u^{2}}{2}) + C [/tex]

    [tex] \frac{1}{4}[\frac {(4x+1)^{4}}{4}+\frac{(4x+1)^{3}}{3}+\frac{(4x+1)^{2}}{2}] +C [/tex]

    So I suppose the only portion that I really don't understand is how they got the 1/4 dx value out of seemingly nothing....
  2. jcsd
  3. Apr 26, 2010 #2
    [tex] du = 4 dx [/tex] solve this for [tex] dx [/tex]
  4. Apr 26, 2010 #3
    I guess I just didn't really think about it all that hard... I got another question though, but Ill post it in a new thread
  5. Apr 26, 2010 #4
    Your solution is correct.
  6. Apr 26, 2010 #5
    There appears to be many notations and definitions of "dy" where y is a dummy variable. Some authors chose to define dy, while others leave it at "no meaning". I came across one interpretation that might help you...

    du/dx = 4 = 4 dx/dx

    So, (1/4) du/dx = dx/dx

    By suppressing the denominator, we get,

    (1/4) du = dx

    Keeping in mind that the denominator is still there. That is, "dy" is shorthand for "dy/dx".

    I like this interpretation because it does not appear to manipulate du/dx like a fraction.

    Hope this helps!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook