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Thread: Integration by Parts - definite integral

  1. #1
    Junior Member symstar's Avatar
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    Integration by Parts - definite integral

    Just evaluate the integral...
    $\displaystyle \int_{0}^{1/2}\sin^{-1}{x} dx$

    I used...
    $\displaystyle u=\sin^{-1}{x}$
    $\displaystyle du=\frac{dx}{\sqrt {1-x^2}}$
    $\displaystyle dv=dx$
    $\displaystyle v=x$

    The problem is that I get stuck with $\displaystyle \int_{0}^{1/2}\frac{xdx}{\sqrt {1-x^2}}$ eventually and I'm not quite sure where to go with it. I thought of maybe doing a sub, but I'm not sure what to sub for.
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  2. #2
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    Substitute $\displaystyle u = 1-x^2$.
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  3. #3
    Junior Member symstar's Avatar
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    Subbing $\displaystyle t=1-x^2$

    $\displaystyle \frac{1}{2} \sin^{-1}{\frac{1}{2}} + \int_{3/4}^{1} \frac{dt}{-2\sqrt{t}}$

    Is this correct so far?
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