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integrate this

x² / (xsinx +cosx)²

$\int \frac{x^{2}}{(x \sin x + \cos x)^{2}} d x = \int (x \sec x) . \frac{x \cos x}{(x \sin x + \cos x)^{2}} d x = (x \sec x) \times \frac{- 1}{(x \sin x + \cos x)} - \int (\sec x + x \sec x \tan x) \times \frac{- 1}{(x \sin x + \cos x)} d x [As \int \frac{x \cos x}{(x \sin x + \cos x)^{2}} d x = \int \frac{d t}{t^{2}} where t = x \sin x + \cos x] = \frac{- x \sec x}{(x \sin x + \cos x)} + \int \sec^{2} x d x = \frac{- x \sec x}{(x \sin x + \cos x)} + \tan x + C = \frac{- x}{\cos x (x \sin x + \cos x)} + \tan x + C = \frac{- x + \sin^{2} x (x \sin x + \cos x)}{\cos x (x \sin x + \cos x)} + C = \frac{- x \cos^{2} x + \sin x \cos x}{\cos x (x \sin x + \cos x)} + C = \frac{(\sin x - x \cos x)}{(x \sin x + \cos x)} + C$

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