The integral ∫ ( x xsinx + x ) ^ 2 dx is equal to, (where C is a...

$∫ \dfrac{{x}^{2}dx}{{\left(xsinx + \cos x\right)}^{2}} = ∫ \dfrac{x}{\cos x} ⋅ \dfrac{xcosx}{{\left(xsinx + \cos x\right)}^{2}}dx$

$= \dfrac{x}{\cos x} ∫ \dfrac{xcosx}{{\left(xsinx + \cos x\right)}^{2}}dx - ∫ \left[\dfrac{d}{dx}\left(xsecx\right) ∫ \dfrac{xcosx}{{\left(xsinx + \cos x\right)}^{2}}dx\right]dx$

$= \dfrac{x}{\cos x}\left(- \dfrac{1}{xsinx + \cos x}\right) + ∫ \sec^{2}xdx$, $\text{∵} ∫ \dfrac{xcosx}{{\left(xsinx + \cos x\right)}^{2}}dx = \dfrac{- 1}{xsinx + \cos x}$

$\text{ and } \dfrac{d}{dx}\left(xsecx\right) = \sec x + xsecxtanx = \sec x\left(1 + \dfrac{xsinx}{\cos x}\right) = \sec^{2}x\left(xsinx + \cos x\right)$

$= \dfrac{- x}{\cos x\left(xsinx + \cos x\right)} + \dfrac{\sin x}{\cos x} + C$

$= \tan x - \dfrac{xsecx}{xsinx + \cos x} + C$