Conisder the following problems $$\int e^x \Big(\frac{1-\sin x}{1-\cos x}\Big)dx.$$ or $$ \int e^x\Big(\frac{1+x\ln x}{x}\Big) dx$$
I have seen these problems (or simpler to them) in many various places but I do not have a specific reference for them. However, I did not see them in the standard calculus books like Stweart unless I missed it. The idea to solve these problems by using the following $$\int e^x (f(x)+f'(x)) dx=\int \frac{d}{dx} \Big(e^x f(x)\Big)dx.$$ Generally speaking, it depends about simplifying the function that are multiplying by $e^x$ and rewrites it as $f(x)+f'(x).$
For the seek of completeness, I will answer the first one by using the rule that I mention. Also, I was curious to see if there is another way to solve them by using one of the integral techniques.
Here is my answer $$\int e^x \Big(\frac{1-\sin x}{1-\cos x}\Big)dx=\int e^x \Big(\frac{1-2 \sin (\frac{x}{2}) \cos (\frac{x}{2})}{2\sin^2(\frac{x}{2})}\Big)dx=\int e^x \Big(\frac{1}{2}\csc^2 (\frac{x}{2})-\cot (\frac{x}{2})\Big) dx, $$ This implies to $$\int e^x \Big(\frac{1-\sin x}{1-\cos x}\Big)dx=\int d \Big(- e^x \cot (\frac{x}{2})+C\Big)) dx= - e^x \cot (\frac{x}{2})+C.$$