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!

Evaluate the integral (inverse trig)

  1. Mar 29, 2010 #1
    1. The problem statement, all variables and given/known data
    Evaluate the following integral:

    2. Relevant equations
    d(arcsinx)/dx = 1/sqrt(1-x2)

    3. The attempt at a solution
    I just need a good start in the right direction on this one so I can at least try it. I know the arcsin d/dx as above, but how do I make this work? Is it some kind of integration by parts or reduction? I have no idea how to go about dealing with the sqrt(2x-x2) mostly, or how it would get there from arcsin...

    any help you can give would be much appreciated!
  2. jcsd
  3. Mar 29, 2010 #2
    Try completing the square.
  4. Mar 29, 2010 #3
    hm.. I'm trying that but still stuck, I get

    int: 1/sqrt(x(2-x))dx, I can move the root up to the top easily enough by multiplying by sqrt(x(2-x))/sqrt(x(2-x)), but that doesn't help much. I found the formula relating a/sqrt(b+x2) to (a/sqrt(b))arctan(x/sqrt(b))+C, but that still doesn't help me with the 2x...
  5. Mar 29, 2010 #4
    No no. -x^2 + 2x = -(x^2 - 2x) = -(ax + b)^2 + c. Find a, b, and c.

    Once you do that, what happens if you let u = ax + b?
  6. Mar 29, 2010 #5
    by that, it should be
    -(x2-2x) = -((x-1)2-1), or a=1, b=-1, c=-1
    so then, making u=x-1, du=1, it becomes

    int{ du/sqrt(-u2-a2)

    and that falls into the arccosh formula as:
    cosh-1(1-x)+c, and that should be the final answer for the indefinate integral right? or can I just plug the -u in like that?
  7. Mar 29, 2010 #6
    Be careful. You are saying -(x-1)^2 - 1 = 2x - x^2?

    What if x = 1? Then, -1 = 2 - 1 = 1, which is wrong. It should be -(x-1)^2 + 1.
    So if we let u = x - 1
    So, we have,
    \int \frac{1}{\sqrt{1 - u^2}} du
    That should look more familiar.
  8. Mar 29, 2010 #7
    ooh, my bad... so its equal to arcsin(x-1)+C !
  9. Mar 29, 2010 #8
    Looks about right. : )
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook