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Question
\[\int \cot^5 \text{ x } {cosec}^4 x\text{ dx }\]
Sum
Solution
∫ 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
\[= - \left[ \frac{t^6}{6} + \frac{t^8}{8} \right] + C\]
\[ = - \left[ \frac{\cot^6 x}{6} + \frac{\cot^8 x}{8} \right] + C\]
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