Evaluate : \(\int\) \(cos^4xdx\over {sin^3x{(sin^5x + cos^5x)^{3\over 5}}}\)

Solution :

I = \(\int\) \(cos^4xdx\over {sin^3x{(sin^5x + cos^5x)^{3\over 5}}}\)

= \(\int\) \(cos^4xdx\over {sin^6x{(1 + cot^5x)^{3\over 5}}}\) = \(\int\) \(cot^4xcosec^2xdx\over {{(1 + cot^5x)^{3\over 5}}}\)

Put \(1+cot^5x\) = t

\(5cot^4xcosec^2x\)dx = -dt

= -\(1\over 5\) \(\int\) \(dt\over {t^{3/5}}\) = -\(1\over 2\) \(t^{2/5}\) + C

= -\(1\over 2\) \({(1+cot^5x)}^{2/5}\) + C