`int(x^(2))/((x sinx+cosx)^(2))dx` is equal to

Options

A. `(sin x+cos x)/(xsin x+cos x)+C`
B. `(x sin x-cos x)/(x sin x+cos x)+C`
C. `(sin x-x cos x)/(x sin x+cos c)+C`
D. None of these

Answer

C

Explanation

Since, `(d)/(dx)(x sin x+cos x)=x cos x`
`therefore I =int (x^(2)dx)/((xsin x+cos x)^(2))`
`=int (x)/(cos x).(x cos x)/(cos x(x sin x+cos x)^(2))dx`
`=(x)/(cos x)((-1)/(x sin x+cos x))`
`-int(cos x-x(-sin x))/(cos ^(2)x)*(-1)/((x sin x+cos x))dx`
`=(-x)/(cos x(x sin x+cos x))+int sec^(2)dx`
`=(-x)/(cos x(xsin x+cos x))+tan x+C`
`=(-x+sin x(x sin x+cos x))/(cos x(x sin x+c os x))+C`
`=(-xcos ^(2)x+sin x*c os x)/(cos x(x sin x+cos x))+C`
`=((sin x-xcos x)/(cos x+xsin x))+C`