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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'](https://en.wikipedia.org/wiki/Separation_of_variables) in order to solve the equation. Once you've done that, the integral will be a standard result.2012-04-29
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    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.2012-04-29
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    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.2012-04-29

2 Answers 2

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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 $$ r = \sin(t + C). $$