Hey everyone. I've been trying to find the length of this parametric curve, but haven't had much luck.

$\displaystyle x=cos{t}$

$\displaystyle y=t+sin{t}$

$\displaystyle 0\leq t \leq \pi$

After finding $\displaystyle \frac{dy}{dt}$ and $\displaystyle \frac{dx}{dt}$ and substituting them into the formula for curve length, I got:

$\displaystyle \int^\pi_0 \sqrt{2+2cost}\ dt$

After this, I tried u-substitution, but couldn't figure out how to eliminate the resultant $\displaystyle t$:

$\displaystyle u = 2+2cost $

$\displaystyle du = -2sint\ dt$

$\displaystyle dt = \frac{du}{-2sint} $

I consulted Wolfram Alpha as a last result, but couldn't figure out how they got the $\displaystyle \int \frac{1}{\sqrt{4-u}}\ du$

Help would be greatly appreciated!