Integrate over $t$ if $dr/dt = \sqrt{1 - r^2}$?

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

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