Use the substitution $u = x\sin x + \cos x$ to find $$ \int \frac{x }{x\tan x + 1 } \, \mathrm{d} x \,. $$ Find by means of a similar substitution, or otherwise, $$ \int \frac{x }{x\cot x - 1 } \, \mathrm{d} x \,. $$
Use a substitution to find $$ \int \frac{x\sec^2 x \, \tan x}{x\sec^2 x - \tan x} \,\mathrm{d} x \, $$ and $$ \int \frac{x\sin x \cos x}{(x - \sin x \cos x)^2} \, \mathrm{d} x \,. $$
