The integral ( x x sin x + cos x ) 2 d x is equal to (where C is a constant integration):

Question

The integral $\int {(\frac{x}{x \sin x + \cos x})}^{2} d x$ is equal to (where C is a constant integration):

Smart Achievers · Accepted Answer

(1)

$I = \int {(\frac{x}{x \sin x + \cos x})}^{2} d x$

$Notice,$

$\frac{d}{d x} (\tan x - \frac{x \sec x}{x \sin x + \cos x}) = \frac{x^{2}}{{(x \sin x + \cos x)}^{2}}$

$Hence,$

$I = \tan x - \frac{x \sec x}{x \sin x + \cos x} + C$

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