Is there a nice solution to this integral: $\int\frac{-a^2 da} {C^2 \sqrt{1-\frac{a^2}{C^2}}}$
Is there an analytical solution to the following integral:
0
$\begingroup$
calculus
-
2You 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
1
Hint
Try substitution $a=C\sin{t}$
2
Take $a=C\sin(\theta)$ so your integral became: $\frac{-1}{C}\int \sin^2(\theta)d\theta$ which is elementary.
1
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]:
1
Yes. For integrals you can always go to wolfram|alpha and they'll tell you what to do. The solution is
$\frac{1}{2} \left(-a \sqrt{1-\frac{a^2}{c^2}}+c \text{ArcSin}\left[\frac{a}{c}\right]\right)$
1
Yes. To solve it you need to do a trig substitution.