The given function is \( \displaystyle \frac{e^{x} \sin x+\cot x+x \sin x}{\sin x} \) and we need to find the integration of this function with respect to \( x \).
Solution:
\( \begin{align*} &\int \frac{e^x \sin x + \cot x + x \sin x}{\sin x} \, dx \\ &= \int \frac{e^{x} \sin x}{\sin x} dx + \int \frac{\cot x}{\sin x} dx + \int \frac{x \sin x}{\sin x} dx \\ &= \int e^{x} dx + \int \cot x \cdot \operatorname{cosec} x dx + \int x dx \\ &= e^{x} -\operatorname{cosec} x + \frac{x^{2}}{2} + C, \quad \text{C is a constant} \\ \end{align*} \)Thus, the integration of the given function is
\( \boxed{\int \frac{e^x \sin x + \cot x + x \sin x}{\sin x} \, dx = e^{x} -\operatorname{cosec} x + \frac{x^{2}}{2} + C} \\ \)Please let me know in the comments if you find any errors in this solution.