<div>Evaluate $$\int \frac{\cos^4 x \, dx}{\sin^3 x \{\sin^5 x + \cos^5 x\}^{3/5}}$$</div>
Step-by-Step Solution
Key Concept: General
<div>$I = \int \frac{\cos^4 x}{\sin^3 x \{\sin^5 x + \cos^5 x\}^{3/5}} \, dx = \int \frac{\cos^4 x}{\sin^6 x \{1 + \cot^5 x\}^{3/5}} \, dx = \int \frac{\cot^4 x \csc^2 x \, dx}{(1 + \cot^5 x)^{3/5}}$<br>Put $1 + \cot^5 x = t \implies 5 \cot^4 x \csc^2 x \, dx = -dt$<br>$\therefore I = -\frac{1}{5} \int \frac{dt}{t^{3/5}} = -\frac{1}{2} t^{2/5} + C = -\frac{1}{2} (1 + \cot^5 x)^{2/5} + C$</div>
Correct Answer: $-\frac{1}{2} (1 + \cot^5 x)^{2/5} + C$