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Math Help - Integrate sqrt(1-x^2)

  1. #1
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    Integrate sqrt(1-x^2)

    I need to integrate sqrt(1-(x^2))

    I know that the reciprical of this integrates to arcsin(x) + k but I don't know how I can relate this.


    Any help would be much appreciated
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  2. #2
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    Re: Integrate sqrt(1-x^2)

    Quote Originally Posted by kinhew93 View Post
    I need to integrate sqrt(1-(x^2))
    I know that the reciprical of this integrates to arcsin(x) + k but I don't know how
    Why do you not use web resources like this?
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  3. #3
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    Re: Integrate sqrt(1-x^2)

    Hello, kinhew93!

    I \;=\; \int \sqrt{1-x^2}\,dx

    It should be obvious that this requires Trig Substitution.

    Let x \,=\,\sin\theta \quad\Rightarrow\quad dx \,=\,\cos\theta\,d\theta

    . . And: . \sqrt{1-x^2} \:=\:\sqrt{1-\sin^2\!x} \:=\:\sqrt{\cos^2\!x} \:=\:\cos x

    Substitute: . I \;=\;\int\cos\theta(\cos\theta\,d\theta) \;=\;\int\cos^2\!\theta\,d\theta \;=\;\tfrac{1}{2}\int (\cos2\theta - 1)\,d\theta

    Can you continue?
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