I'm stuck on this step trying to solve a math puzzle. How do I integrate this?
Integrate over $t$ if $dr/dt = \sqrt{1 - r^2}$?
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$\begingroup$
calculus
integration
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0you 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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0Stuck 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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0Please 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
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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 $$ r = \sin(t + C). $$