If ∫ dx ^ 3 x 2 2x = ( x) ^ A + C ( x) ^ B + k , where k is a...

$∫ \dfrac{dx}{\cos^{3}x\sqrt{4\sin x \cos x}} = ∫ \dfrac{dx}{2\cos^{4}x\sqrt{\tan x}}$

Let, $\tan x = {t}^{2} ⇒ \sec^{2}x = 1 + {t}^{4}$

And $\sec^{2}xdx = 2tdt$

$∴ I = ∫ \dfrac{\sec^{4}xdx}{2\sqrt{\tan x}} = ∫ \dfrac{\sec^{2}x . \sec^{2}xdx}{2\sqrt{\tan x}}$

$= ∫ \dfrac{\left(1 + {t}^{4}\right)2tdt}{2t} = ∫ \left(1 + {t}^{4}\right)dt$

$= t + \dfrac{{t}^{5}}{5} + k$, where $k$ is a constant of integration.

$= \sqrt{\tan x} + \dfrac{1}{5}tan\dfrac{5}{2}x + k$

Where, $t = \sqrt{\tan x}$.

$A = \dfrac{1}{2}, B = \dfrac{5}{2}, C = \dfrac{1}{5}$

$A + B + C = \dfrac{16}{5}$