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Math Help - Evaluating this indefinite integral involving arc sin

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    Evaluating this indefinite integral involving arc sin

    The indefinite integral of sin^-1(x) or arc sin (x) divided by the square root of 1-x^2. All I can say is I have no idea what to do with this, I've never really worked with arcsin, I looked up derivatives and it doesn't seem like I can make one part of the equation's derivative cancel out another so I don't know if substitution would be a good route here. Could somebody help me out???
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by fattydq View Post
    The indefinite integral of sin^-1(x) or arc sin (x) divided by the square root of 1-x^2. All I can say is I have no idea what to do with this, I've never really worked with arcsin, I looked up derivatives and it doesn't seem like I can make one part of the equation's derivative cancel out another so I don't know if substitution would be a good route here. Could somebody help me out???
    If you're dealing with \int\frac{\sin^{-1}x}{\sqrt{1-x^2}}\,dx, make the substitution u=\sin^{-1}x. Things will work out really nice from there.

    Can you continue?
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    So wouldn't I just be left with the indefinite integral of u? But then how does that simplify? Because the antideriv. would be u^2/2 so my final answer would be (sin^-1(x))2/2? Or am I mistaken?
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by fattydq View Post
    So wouldn't I just be left with the indefinite integral of u? But then how does that simplify? Because the antideriv. would be u^2/2 so my final answer would be (sin^-1(x))2/2? Or am I mistaken?
    You're not mistaken. The answer would be \frac{\left(\sin^{-1}x\right)^2}{2}+{\color{red}C}
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    Quote Originally Posted by fattydq View Post
    So wouldn't I just be left with the indefinite integral of u? But then how does that simplify? Because the antideriv. would be u^2/2 so my final answer would be (sin^-1(x))2/2? Or am I mistaken?
    Yes, you've got it (assuming the first two is an exponent). Don't forget the constant.
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