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Integration by trigonometric change of variable

  1. Aug 1, 2014 #1

    MMM

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    1. The problem statement, all variables and given/known data
    I'm trying to solve ##\int\sqrt{a^2 - x^2}##
    by using the substitution ##x = asin\theta##

    2. Relevant equations

    ##x = asin\theta

    3. The attempt at a solution
    ##y = \int\sqrt{a^2 - a^2cos^2\theta}##
    ##y = a\int\cos\theta##
    ##y = a^2\int\cos(\theta)^2##
    ##y = (a^2)/2 * \int1+cos2\theta##
    ##y = a^2/2 + a^2/4 * sin2\theta\ + C##
    ##\theta = arcsin(x/a)##
    ##a^2/2 * arcsin(x/a) + a^2/2 * sin\theta\cos\theta\ + C## Any help would greatly be appreciated.

    The answer in the book is ##(a^2/2) * arcsin(x/a) + (x/2) * \sqrt{a^2 - x^2} + C##

    EDIT: I figured it out.
     
    Last edited: Aug 1, 2014
  2. jcsd
  3. Aug 1, 2014 #2

    verty

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    Homework Helper

    You have some crazy algebra happening there. You are not ready for tests or exams yet because your working out is incorrect. It must be neat and each statement must follow from the previous one. For example, this is meaningless: ##y = a \int cos\theta##, that is not how an integral is written.
     
  4. Aug 2, 2014 #3

    HallsofIvy

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    Staff Emeritus
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    You have started off by copying the problem wrong. It should be ##\int\sqrt{a^2- x^2}dx##
    Do you see the difference?

    Same mistake as before.

    I have not idea where you got this, [tex]\sqrt{a^2- a^2 cos(\theta)} is NOT equal to "[itex]a cos(\theta)[/itex]"

    And this is definitely not equal to the previous line!

     
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