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Evaluate using any method Number 2

  1. Aug 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Evaluate using any method:

    [tex]\int\frac{8x^3+10} {\sqrt[3]{5x-4}}[/tex]dx



    2. Relevant equations



    3. The attempt at a solution

    I'm lost on this one.
     
    Last edited: Aug 2, 2010
  2. jcsd
  3. Aug 2, 2010 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Since it is that cube root in the denominator that is causing the trouble, a kind of obvious substitution is [itex]u= 5x- 4[/itex]. Then du= 5dx or dx= (1/5)du.

    Also, 5x= u+ 4 so x= (u+ 4)/5. Replacing x by that in [itex]8x^3+ 10[/itex] gives you a cubic, in u, in the numerator divided by [itex]\sqrt[3]{u}= u^{1/3}[/itex]. That reduces to a sum of terms involving [itex]u^{3- 1/3}= u^{8/3}[/itex], [itex]u^{2- 1/3}= u^{5/3}[/itex], and [itex]u^{1- 1/3}= u^{-2/3}[/itex].
     
  4. Aug 2, 2010 #3
    Where do you get the 5x=u+4?
     
  5. Aug 2, 2010 #4

    Mark44

    Staff: Mentor

    From the substitution u = 5x - 4.
     
  6. Aug 2, 2010 #5
    So do you get something like:

    8[tex]\int\frac{(u+4/5)^3 + 10}{u^(1/3)}[/tex]

    That's U raised to the 1/3 on the denominator
    ?
     
  7. Aug 2, 2010 #6

    Mark44

    Staff: Mentor

    It's not u + 4/5 in the numerator -- it's (1/5)(u + 4). Also, where is du in your integral? You seem to be ignoring it.
     
  8. Aug 2, 2010 #7
    Yea thats what I meant. I don't understand where the Du goes.

    8[tex]\int\frac{((u+4)/5)^3 + 10}{u^(1/3)}[/tex]

    So that is right? Then does du takes the spot of the 5?
     
  9. Aug 2, 2010 #8

    hunt_mat

    User Avatar
    Homework Helper

    Use u=5x-4 then du=5dx, insert this into the integral to obtain:
    [tex]
    \int\frac{8x^3+10} {\sqrt[3]{5x-4}}dx=\frac{1}{5}\int\frac{8((u+4)/5)^{3}+10}{u^{1/3}}du
    [/tex]
    Now all you have to do is expand the cube and integrate using
    [tex]
    \int x^{n}dx=\frac{1}{n+1}x^{n+1}+c
    [/tex]
     
  10. Aug 2, 2010 #9
    What is n?
     
  11. Aug 2, 2010 #10

    hunt_mat

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    Homework Helper

    [tex]n[/tex] is a number, like 3 or 2/3.
     
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