Let \(I = \int \left(\frac{x}{x\sin x + \cos x}\right)^2 dx \)
\(= \int \frac{x^2 }{(x \sin x + \cos x)^2}dx\)
\(= \int \frac x{\cos x}. \frac {x \cos x}{(x \sin x + \cos x)^2} dx\)
Solving by integration by parts,
\(I = \frac x {\cos x}\left( \frac{-1}{x\sin x + \cos x}\right) + \int \left(\frac {\cos x + x\sin x}{\cos^2 x}\right)\left(\frac 1{x\sin x + \cos x}\right)dx\)
\(= -\frac{x \sec x}{x \sin x + \cos x} + \int \sec^2x dx\)
\(= -\frac{x\sec x}{x \sin x + \cos x} +\tan x + C\)
