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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 $a$.2012-09-27

5 Answers 5

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Hint

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

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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) Out[3]:         ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽                 ╱    2                  x⋅╲╱  - x  + 1    asin(x) - ─────────────── + ───────          2             2     In [4]:  
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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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Yes. To solve it you need to do a trig substitution.