Question:

Let $I = \int \frac{\sin^{1/2} x}{\cos^{9/2} x} dx = \int \frac{dx}{\sin^{-1/2} x \cos^{9/2} x}$<br>Here $m + n = \frac{1}{2} - \frac{9}{2} = -4$ (negative even integer).<br>Divide Numerator & Denominator by $\cos^4 x$.<br>$I = \int \sqrt{\tan x} \sec^4 x dx = \int \sqrt{\tan x} (1 + \tan^2 x) \sec^2 x dx$<br>$= \int \sqrt{t} (1 + t^2) dt$ (using $\tan x = t$)<br>$= \frac{2}{3} t^{3/2} + \frac{2}{7} t^{7/2} + C = \frac{2}{3} \tan^{3/2} x + \frac{2}{7} \tan^{7/2} x + C$