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: Find a and b

  1. Jun 25, 2013 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    If [itex] \displaystyle \int^b_a \dfrac{x^n dx}{x^n + (16-x)^n} = 6 [/itex] and a+b=16, then find a and b.

    3. The attempt at a solution

    [itex] \displaystyle \int^b_a \dfrac{dx}{1 + (16/x - 1)^n} = 6 [/itex]

    I tried substitution but it did not work.
  2. jcsd
  3. Jun 25, 2013 #2


    User Avatar
    Homework Helper

    What sub did you try? Hint: try ##x = 16-y## on the original integral.

    This is a simple algebra problem, more or less. Very little actual integration need be done.
  4. Jun 25, 2013 #3


    User Avatar
    Homework Helper

    \int_a^b\frac{x^n}{x^n + (16-x)^n}\,\mathrm{d}x = \int_a^b 1\,\mathrm{d}x - \int_a^b\frac{(16-x)^n}{x^n + (16-x)^n}\,\mathrm{d}x
    Can you find a substitution which turns the integrand of the second integral on the right into the integrand of the integral on the left?
  5. Jun 26, 2013 #4
    Why do you need substitution right now ?

    Apply the property:

    I= [itex] \displaystyle \int^b_a \dfrac{x^n dx}{x^n + (16-x)^n} [/itex]

    And I= [itex] \displaystyle \int^b_a \dfrac{(16-x)^n dx}{(16-x)^n + (16-(16-x))^n}[/itex]

    Now add the two to get,

    2I= ab∫dx

    Can you proceed? You already know that I=6...
  6. Jun 26, 2013 #5


    User Avatar
    Gold Member

    Oh that was so easy. I first thought of applying property but somehow couldn't notice that a+b=16 was already given.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted