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Math Help - Another seemingly simple integral

  1. #1
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    Another seemingly simple integral

    The definite integral from -1/2 to 1/2 of 9/the square root of 1-x^2 dx

    I don't even know where to begin in this one. Finding the antiderivative of that seems impossible...
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  2. #2
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    \int_{-1/2}^{1/2}{\frac{9}{\sqrt{1-x^{2}}}\,dx}=2\int_{0}^{1/2}{\frac{9}{\sqrt{1-x^{2}}}\,dx}.

    Look at a table of integrals, that primitive is the arcsin function.
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  3. #3
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    Quote Originally Posted by Krizalid View Post
    \int_{-1/2}^{1/2}{\frac{9}{\sqrt{1-x^{2}}}\,dx}=2\int_{0}^{1/2}{\frac{9}{\sqrt{1-x^{2}}}\,dx}.

    Look at a table of integrals, that primitive is the arcsin function.
    I know that, but what did you just do knowing that? Where did the 2 come from and why did the -1/2 just disappear? And I know that 1/the sq root of 1-x^2 but it's a 9 not a 1, so where does this come into play?
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  4. #4
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    \int_{-a}^af=2\int_0^af provided that f is an even function and a\in\mathbb R.
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  5. #5
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    Quote Originally Posted by fattydq View Post
    I know that, but what did you just do knowing that? Where did the 2 come from and why did the -1/2 just disappear? And I know that 1/the sq root of 1-x^2 but it's a 9 not a 1, so where does this come into play?
    \int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{9}{\sqrt{1-x^2}}dx=<br />
\int_{-\frac{1}{2}}^{0}\frac{9}{\sqrt{1-x^2}}dx+\int_{0}^{\frac{1}{2}}\frac{9}{\sqrt{1-x^2}}dx,
    but \frac{9}{\sqrt{1-(-x)^2}}=\frac{9}{1-x^2}\Rightarrow  \int_{-\frac{1}{2}}^{0}\frac{9}{\sqrt{1-x^2}}dx=\int_{0}^{\frac{1}{2}}\frac{9}{\sqrt{1-x^2}}dx.
    Therefore \int_{-\frac{1}{2}}^{\frac{1}{2}}\frac{9}{\sqrt{1-x^2}}dx=<br />
2\int_{0}^{\frac{1}{2}}\frac{9}{\sqrt{1-x^2}}dx=18\int_{0}^{\frac{1}{2}}\frac{dx}{\sqrt{1-x^2}}.


    PD: Late >.<
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  6. #6
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    Quote Originally Posted by Krizalid View Post
    \int_{-a}^af=2\int_0^af provided that f is an even function and a\in\mathbb R.
    I love calculus, because there's rule's for everything that I'm just supposed to know magically, intuitively. I swear my professor never mentioned anything like that.
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