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Inverse substitution question

  1. Mar 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Let's say you make the inverse substitution x = 2tan(z) in some integral.

    Let's say you evaluate the integral get something like like 4sin(z). How do you put z back in terms of x?

    2. Relevant equations

    3. The attempt at a solution
    I can do it by saying arctan(x/2) = z, but my teacher said we are not allowed to state the answer that way. I'm not sure how else you would do that...?
  2. jcsd
  3. Mar 15, 2009 #2


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    Draw a right triangle where tan(x)=x/2. For example, let the side opposite the angle z be x and the side adjacent be 2. Now use Pythagoras to find the hypotenuse. What then is sin(z) in terms of x? It's opposite over hypotenuse, right?
  4. Mar 15, 2009 #3
    Ohh okay, so in this case you would end up with 4x / sqrt(x^2 + 4).

  5. Mar 15, 2009 #4


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    Exactly. 4*sin(arctan(x/2))=4x/sqrt(x^2+4). Drawing a triangle is nice way to derive stuff like that without memorizing it.
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