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Is there a nice solution to this integral: $$\int\frac{-a^2 da} {C^2 \sqrt{1-\frac{a^2}{C^2}}}$$

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You can use the trigonometric substitution $a = C \sin{\theta}$, $da = C \cos{\theta} \, d\theta$. However, you need to have limits of integration because your integrand is not defined for all values of $a$. – Christopher A. Wong Sep 27 '12 at 7:34
up vote 1 down vote accepted

Yes. For integrals you can always go to wolfram|alpha and they'll tell you what to do. The solution is

$$\frac{1}{2} \left(-a \sqrt{1-\frac{a^2}{c^2}}+c \text{ArcSin}\left[\frac{a}{c}\right]\right)$$

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Take $a=C\sin(\theta)$ so your integral became: $$\frac{-1}{C}\int \sin^2(\theta)d\theta$$ which is elementary.

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Yes. You should try some trigonometric substitution (or install sympy, then can you answer by yourself. A related (simplified= integral:

In [3]: integrate( x**2/sqrt(1-x**2), x)
      ╱    2               
  x⋅╲╱  - x  + 1    asin(x)
- ─────────────── + ───────
         2             2   

In [4]: 
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Yes. To solve it you need to do a trig substitution.

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Try substitution $a=C\sin{t}$

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