Is there an analytical solution to the following integral:

Question

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

Answer 1

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

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

Answer 2

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

Answer 3

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]: 

Answer 4

Yes. To solve it you need to do a trig substitution.

Answer 5

Hint

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