The given function is \(\displaystyle \frac{\csc x}{\csc x \,- \cot x} \) and we need to compute the integral of this function with respect to \( x \), i.e., compute \(\displaystyle \int \frac{\csc x}{\csc x \,- \cot x} \, dx \).
Solution:
Let’s first simplify the Integrand:
\( \begin{aligned} &\frac{\csc x}{\csc x \,- \cot x} \\ &= \frac{\frac{1}{\sin x}}{\frac{1}{\sin x} – \frac{\cos x}{\sin x}} = \frac{\frac{1}{\sin x}}{\frac{1 – \cos x}{\sin x}} \\ &= \frac{1}{1 – \cos x} = \frac{1}{1 – \cos x} \cdot \frac{1 + \cos x}{1 + \cos x} \\ &= \frac{1 + \cos x}{1 – \cos^2 x} = \frac{1 + \cos x}{\sin^2 x} \\ &= \frac{1}{\sin^2 x} + \frac{\cos x}{\sin^2 x} \\ &= \csc^2 x + \csc x \cot x \end{aligned} \)Thus, the integral becomes:
\( \begin{aligned} &\int \left( \csc^2 x + \csc x \cot x \right) dx \\ &= \int \csc^2 x \, dx + \int \csc x \cot x \, dx \\ &= -\cot x + (-\csc x) + C, \quad \text{C is a constant} \\ \end{aligned} \)Thus, the integral of the given function is
\( \boxed{ \int \frac{\csc x}{\csc x \,- \cot x} \, dx = -\cot x \, – \csc x + C } \)Please let me know in the comments if you find any errors in this solution.