Question

\[\int \cot^5 \text{ x } {cosec}^4 x\text{ dx }\]

Answer 1

∫ cot⁵ x . cosec⁴ x dx
= ∫ cot⁵ x . cosec² x . cosec² x dx

= ​​∫ cot⁵ x . (1 + cot² x) . ​cosec² x dx
Let cot x = t

⇒  – cosec² x dx = dt
⇒ cosec² x dx = –dt

Now, ∫ cot⁵ x . cosec⁴ x dx
= ∫ t⁵ (1 + t²) dt
= ∫(t⁵ + t⁷) dt

\[= - \left[ \frac{t^6}{6} + \frac{t^8}{8} \right] + C\]
\[ = - \left[ \frac{\cot^6 x}{6} + \frac{\cot^8 x}{8} \right] + C\]