The value of ∫ cos x ^ 2 x(sin x + cos x) dx is equal to

Let, $I = ∫ \dfrac{cos x}{\sin^{2} x\left(sin x + cos x\right)}dx$

On dividing numerator and denominator by $\cos^{3} x,$ and substituting $\tan x = t, \sec^{2}xdx = dt$, we get,

$I = ∫ \dfrac{dt}{t^{2}\left(1 + t\right)}$$<br /><br />$= ∫ \left[- \dfrac{1}{t} + \dfrac{1}{t^{2}} + \dfrac{1}{1 + t}\right]dt$(using partial fraction)<br /><br />$⇒ I = - ∫ \dfrac{1}{t} dt + ∫ t^{- 2} dt + ∫ \dfrac{1}{1 + t} dt$<br /><br />$= - \log \left|t\right| - \dfrac{1}{t} + \log \left|1 + t\right| + C$<br /><br />$= - \dfrac{1}{tan x} + \log \left|\dfrac{1 + tan x}{tan x}\right| + C$<br /><br />$= - cot x + \log \left|\dfrac{1 + tan x}{tan x}\right| + C