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A clarification on a step in an integration question

  1. Jan 16, 2016 #1
    1. The problem statement, all variables and given/known data

    I was given this question as a part of an assignment and lost a mark because of a step.


    2. Relevant equations
    the integral of
    cos^5(x) dx

    after some fiddling and substitution it gets to this

    (1 - u^2)^2 du
    In the solutions there is a step that says
    refine
    = (u^2 - 1)^2
    basically switching the the 1 and the u^2 around.

    3. The attempt at a solution
    Is this possible and if so how?
     
  2. jcsd
  3. Jan 16, 2016 #2

    BvU

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    It is possible because ##(-1)^2 = 1## and the distributive property of multiplication:$$ \eqalign { (a^2-1)^2 & = 1 * (a^2-1)^2 \\ & = (-1)^2 * (a^2-1)^2 \\ & = ( -1*(a^2-1) ) * ( -1*(a^2-1) ) \\ & = ( -1*(a^2-1) )^2 \\ & = ( 1 - a^2 )^2 } $$

    Or, simply, because ## a^2 = (-a)^2 ##...:rolleyes:
     
  4. Jan 16, 2016 #3
    Good point. I'll have to remember that.
     
  5. Jan 16, 2016 #4

    Ray Vickson

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    I don't know why you lost a mark. Both (1-u^2)^2 and (u^2 - 1)^2 are equal to u^4 - 2 u^2 + 1, so IF you performed that expansion you should not have lost a mark.
     
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