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Math Help - Integration with a squareroot

  1. #1
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    Integration with a squareroot

    Could someone please help me solve this integral: I = squareroot(1-x^2) dx
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  2. #2
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    Re: Integration with a squareroot

    x=sin(t)
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  3. #3
    MHF Contributor ebaines's Avatar
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    Re: Integration with a squareroot

    Start by substituting x = sin(w), so that dx = cos(w) dw. You'll end up having to integrate \cos^2(x)dx, and to do that you can use the half angle formula: \cos^2w = \frac 1 2 (1+\cos(2w)). Try it, and post back with what you get.
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  4. #4
    Senior Member x3bnm's Avatar
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    Re: Integration with a squareroot

    You may want to check this page which has detailed instructions(and demonstration) on how to integrate \int \sqrt{1 - x^2}\,\,dx as a definite integral. For indefinite integral the procedure is same.

    definite integrals
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  5. #5
    Senior Member x3bnm's Avatar
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    Re: Integration with a squareroot

    To make it easier I re-edited the post. We have to evaluate:
    \int \sqrt{1-x^2}\,\,dx

    Here it is again:

    Let's think of x and 1 in the above question as two sides of a right triangle.

    Integration with a squareroot-triangle.png


    \cos{(\theta)} = \frac{x}{1} = x

    \sqrt{1-x^2} = \sqrt{1-\cos^2{(\theta)}} = \sin{(\theta)}

    \text{And } dx = -\sin{(\theta)}\,\,d\theta


    \begin{align*}\int \sqrt{1-x^2}\,\,dx =& -\int \sin^2{(\theta)}\,\,d\theta \\=& \int \frac{\cos{(2\theta)}-1}{2}\,\,d\theta....\text{[Using trigonometry]} \\=& \frac{\sin{(2\theta)}}{4} -\frac{\theta}{2} + C \\=& \frac{\sin{(\theta)}\cos{(\theta)}}{2} - \frac{\theta}{2} + C......\text{[using trigonometry]}\\=& \frac{x\sqrt{1-x^2}}{2} - \frac{\cos^{-1}{(x)}}{2}+ C\\& \text{.....[by replacing } \sin{(\theta)}, \cos{(\theta)} \text{ and } \theta \text{]}\end{align*}

    Hope it helps.
    Last edited by x3bnm; April 23rd 2013 at 12:28 PM.
    Thanks from jdesak
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  6. #6
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    Re: Integration with a squareroot

    Note that either x= sin(w), as suggested by ebaines, or [itex]x= sin(\theta)[/itex], as suggested by X3bnm, will work in exactly the same way.
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