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I'm stuck on this step trying to solve a math puzzle. How do I integrate this?

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you cannot integrate this directly but need to first 'separate the variables' in order to solve the equation. Once you've done that, the integral will be a standard result. – Ronald Apr 29 '12 at 23:07
Stuck on what step? Please make the body of your post self-contained. You don't ask readers of a book to start at the spine; you shouldn't ask readers of your post to start at the subject, which is just an indexing item. – Arturo Magidin Apr 29 '12 at 23:07
Please include the question in the question, not just in the title. In any event, it is not clear exactly what integral you are trying to evaluate. – Gerry Myerson Apr 29 '12 at 23:07
up vote 2 down vote accepted

You have that

$$\frac{{dr}}{{dt}} = \sqrt {1 - {r^2}} $$

from where

$$\frac{{dr}}{{\sqrt {1 - {r^2}} }} = dt$$

then integrating

$$\arcsin r + C = t$$

You need a initital value to determine $C$.

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I think you are asking to solve a differential equation. This one is separable, so one has $$ \arcsin(r) = t + C $$ or $$ r = \sin(t + C). $$

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