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The integral \int \left ( \frac{x}{x \sin +\cos x} \right )^{2}dx is equal to  (where C is constant of integration):
Option: 1 \tan x - \frac{x \sec x}{x \sin x + \cos x}+C
Option: 2 \sec x \frac{x \tan x}{x \sin x + \cos x}+C
Option: 3 \sec x - \frac{x \tan x}{x \sin x + \cos x}+C
Option: 4 \tan x + \frac{x \sec x}{x \sin x+ \cos x}+C

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\int\left(\frac{x}{x \sin x+\cos x}\right)^{2} d x=\int\left(\frac{x}{\cos x}\right) \cdot \frac{x \cos x d x}{(x \sin x+\cos x)^{2}}

\\=\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) d 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

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himanshu.meshram

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