1. We need to integrate the function $$\int \csc^5 x \, dx$$. 2. Express $$\csc^5 x$$ as $$\csc^3 x \cdot \csc^2 x$$ to use identities. 3. Recall that $$\csc^2 x = 1 + \cot^2 x$$, so rewrite the integral as $$\int \csc^3 x (1 + \cot^2 x) \, dx$$. 4. Use substitution: let $$u = \cot x$$, then $$du = -\csc^2 x\, dx$$ or $$-du = \csc^2 x \, dx$$. 5. However, to handle $$\csc^3 x$$, write it as $$\csc x \cdot \csc^2 x$$, so the integral becomes $$\int \csc x \cdot \csc^2 x (1 + \cot^2 x) \, dx = \int \csc x (1 + \cot^2 x) \csc^2 x \, dx$$. 6. Substitute $$\csc^2 x \ dx = -du$$ (noting careful rearrangement), then express entire integral in terms of $$u$$. 7. A better approach is to write $$\csc^5 x = \csc^3 x \csc^2 x$$ and write $$\csc^3 x = \csc x \csc^2 x$$. 8. Using this, the integral is $$\int \csc x \csc^2 x \csc^2 x \, dx = \int \csc x (\csc^2 x)^2 \, dx$$, which is complicated. 9. Alternatively, use the power-reduction formula for cosecant powers or express $$\csc^5 x$$ as $$\csc^3 x \csc^2 x$$ and use substitution for $$\cot x$$. 10. Express $$\csc^5 x = \csc^3 x \csc^2 x$$ and use substitution $$u = \cot x$$ with $$du = -\csc^2 x \, dx$$, so $$dx = -\frac{du}{\csc^2 x}$$. 11. Substitute back into the integral: $$\int \csc^3 x \csc^2 x \, dx = \int \csc^3 x \cdot \csc^2 x \, dx = \int \csc^3 x ( -du / \csc^2 x ) = - \int \csc x \, du$$. 12. Express $$\csc x$$ in terms of $$u = \cot x$$: Recall $$\csc^2 x = 1 + u^2$$, so $$\csc x = \sqrt{1 + u^2}$$. 13. Therefore, integral becomes $$- \int \sqrt{1 + u^2} \, du$$. 14. Now we integrate $$- \int \sqrt{1 + u^2} \, du$$. 15. This is a standard integral with formula: $$\int \sqrt{a^2 + x^2} \, dx = \frac{x}{2} \sqrt{a^2 + x^2} + \frac{a^2}{2} \ln |x + \sqrt{a^2 + x^2}| + C$$ with $$a=1$$. 16. Applying the formula: $$- \left( \frac{u}{2} \sqrt{1 + u^2} + \frac{1}{2} \ln |u + \sqrt{1 + u^2}| \right) + C$$. 17. Substitute back $$u = \cot x$$: $$- \frac{\cot x}{2} \sqrt{1 + \cot^2 x} - \frac{1}{2} \ln |\cot x + \sqrt{1 + \cot^2 x}| + C$$. 18. Since $$\sqrt{1 + \cot^2 x} = \csc x$$, simplify: $$- \frac{\cot x \csc x}{2} - \frac{1}{2} \ln |\cot x + \csc x| + C$$. 19. Final answer is: $$\boxed{- \frac{\cot x \csc x}{2} - \frac{1}{2} \ln |\cot x + \csc x| + C}$$.