(a) Find: ∫ (x - sin x)/(1- cos x) d x

$\displaystyle =\int \dfrac{x}{1-\cos x} d x-\int \dfrac{\sin x}{1-\cos x} d x$

$\displaystyle =\int \dfrac{x}{2 \sin^{2} \frac{x}{2}} d x-\int \dfrac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \sin^{2} \frac{x}{2}} d x$

$\displaystyle =-\dfrac{x}{2} \cdot 2 \cot \dfrac{x}{2}-\dfrac{1}{2} \int-2 \cot \dfrac{x}{2} d x-\int \cot \dfrac{x}{2} d x$

$\displaystyle =-x \cot \dfrac{x}{2}+C$

$\displaystyle =\int_{0}^{2} \dfrac{1}{\sqrt{(x+1)^{2}+2}} d x$

$\displaystyle =\left[\log \left(x+1+\sqrt{x^{2}+2 x+3}\right)\right]_{0}^{2}$

$\displaystyle =\log (3+\sqrt{11})-\log (1+\sqrt{3})$ or $\displaystyle \log\left(\dfrac{3+\sqrt{11}}{1+\sqrt{3}}\right)$