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Evaluate the Improper integral [4,13] 1/(x-5)^(1/3)

  1. Apr 12, 2015 #1
    1. The problem statement, all variables and given/known data

    Evaluate the Improper integral [4,13] 1/(x-5)1/3

    2. Relevant equations

    N/A

    3. The attempt at a solution
    IMG_3094.JPG
    In step 1, I split the integral into two separate integrals because at x=5, it would be undefined. I made the first limit approach 5 from the left and the second limit approach five from the right.

    For step 2, I did u substitution to prepare to integrate. Since I did the substitution, I plugged the old intervals into u = x - 5 to get the new ones.

    For step 3, I rewrote both integrals in terms of u du and brought the u^(1/3) from the denominator to the numerator for easier integration. I also added the new intervals in.

    In step 4, I integrated both integrals.

    Step 5 is where I'm not sure what to do. I plugged in the values and that is what I ended up with but in the first parentheses, the second value has a (-1)2/3 which is undefined. The (R-5)'s will equal 0 and cancel those fractions out but I don't know how to handle the (-1)2/3.
     
  2. jcsd
  3. Apr 12, 2015 #2

    Mark44

    Staff: Mentor

    ##(-1)^{2/3}## is defined -- its value is 1.

    You can work with it either as ##[(-1)^2]^{1/3} = 1^{1/3} = 1## or as ##[(-1)^{1/3}]^2 = (-1)^2 = 1##.
     
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