<s>[INST] Calculating the Integral for the Brachistochrone Problem I am trying to prove that the shortest possible time of descent in the Brachistochrone problem is:
$$ T = \sqrt\frac{c_1}{2g} \theta_1 $$
I have the following 2 parametric equations:
$$ x = \frac{c_1}{2} (\theta - \sin\theta) \qquad y = \frac{c_1}{2} (1 - \cos\theta) $$
Solving $$ \frac{dx}{d\theta} = \frac{c_1}{2} (1-\cos\theta) \qquad \frac{dy}{d\theta} = \frac{c_1}{2} (\sin\theta) $$
I know that $$ T = \int_0^{x_1} \frac{\sqrt{1 + (y')^2}}{\sqrt{2gy}} dx  \tag{1} $$
Now solve $$y' = \frac{dy}{dx} = \frac{dy / d\theta}{ dx / d\theta } = \frac{\sin\theta}{1-\cos\theta} $$
Therefore $$ (y')^2  = \frac{\sin^2\theta}{(1-\cos\theta)^2} = \frac{1+\cos\theta}{1-\cos\theta} $$
Now plugging everything back into \ref{1} and putting the integral in terms of $\theta$:
$$T = \int_0^{\theta_1} \frac{\sqrt{1 + \frac{1+\cos\theta}{1-\cos\theta}}}{\sqrt{2g\frac{c_1}{2} (1 - \cos\theta)}} \frac{c_1}{2} (1-\cos\theta) d\theta$$
I tried solving this integral but I am having a hard time with it. I would appreciate some guidance on how to tackle this integral.
 [/INST] We can work with dimensionless variable, $\large X = {2x \over c_1}\;Y = {2y \over c_1}$
$X = θ - \sin(θ)$
$Y = 1 - \cos(θ)$
${dX \over dθ} = 1 - \cos(θ) = Y$
${dY \over dθ} = \sin(θ) = ±\sqrt{1-\cos^2(θ)} = ±\sqrt{1-(1-Y)^2} = ±\sqrt{2Y-Y^2}$
$\displaystyle \left({dX \over dθ}\right)^2 + \left({dY \over dθ}\right)^2 = Y^2 + (2Y-Y^2) = 2Y$
$\displaystyle T = \int_0^{X_1} \sqrt{1+\left({dY \over dX}\right)^2 \over c_1\;g\;Y} \left( {c_1\over2} dX\right) 
= \sqrt{c_1 \over 4g} \int_0^{θ_1} \sqrt{2Y\over Y} dθ
= \sqrt{c_1 \over 2g}\; θ_1
$
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