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!

Homework Help: Just an integral

  1. Feb 15, 2010 #1
    1. The problem statement, all variables and given/known data

    [tex]\int\frac{dx}{1+\sqrt[3]{x-2}}[/tex]

    2. Relevant equations

    The answer is given: [tex]\frac{3}{2}(x-2)^\frac{3}{2}-3(x-2)^\frac{1}{3}+ln|1+(x-2)^\frac{1}{3}|+C[/tex]

    I have to get my answer to look just like this.

    3. The attempt at a solution

    [tex]u=x-2[/tex], [tex]du=dx[/tex]

    [tex]=\int\frac{dx}{1+\sqrt[3]{u}}[/tex]

    [tex]w=1+\sqrt[3]{u}[/tex]

    [tex]dw=\frac{du}{3u^\frac{2}{3}}[/tex]

    [tex]3u^\frac{2}{3}dw=du[/tex]

    [tex](w-1)^2=u^\frac{2}{3}[/tex]

    [tex]3(w-1)^2dw=du[/tex]

    [tex]=3\int\frac{(w-1)^2dw}{w}[/tex]

    [tex]=3\int\frac{w^2-2w+1}{w}dw[/tex]

    [tex]=3\int\frac{w^2}{w}dw-3\int\frac{2w}{w}dw+3\int\frac{1}{w}dw[/tex]

    [tex]=3\int\(wdw-6\int\(dw+3\int\frac{dw}{w}[/tex]

    [tex]=\frac{3}{2}(w^2)-6w+3ln|w|+C[/tex]

    [tex]=\frac{3}{2}(1+\sqrt[3]{u})^2-6(1+\sqrt[3]{u})+3ln|1+\sqrt[3]{u}|+C[/tex]

    [tex]=\frac{3}{2}(1+\sqrt[3]{x-2})^2-6(1+\sqrt[3]{x-2})+3ln|1+\sqrt[3]{x-2}|+C[/tex]

    From here I don't know where to go to get the answer stated above or if I'm not on the right track.
     
  2. jcsd
  3. Feb 15, 2010 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You are doing fine. Now, just multiply the first two terms out. You can throw the constants out, since you already have a '+C'. Now combine what's left.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for integral Date
Integration substitution rule Saturday at 8:35 AM
Order of multi-variable integration of infinite range Friday at 11:33 PM
Integration over a ball Thursday at 10:31 PM
Fourier transform of integral e^-a|x| Thursday at 8:15 AM
Cylindrical Coordinates Triple Integral -- stuck in one place Wednesday at 9:00 PM