Scary Integration!

Calculus Level 4

x2(xsinx+cosx)2dx\large \displaystyle \int \frac { { x }^{ 2 } }{ ({ x\sin { x } +\cos { x } ) }^{ 2 } } dx

Which of the choices is equivalent to the indefinite integral above (ignoring arbitrary constant)?

A.I=sinx+xcosxxsinxcosx\displaystyle A.\quad I= \frac { \sin { x } +x\cos { x } }{ x\sin { x } -\cos { x } }

B.I=sinxxcosxxsinx+cosx\displaystyle B.\quad I= \frac { \sin { x } -x\cos { x } }{ x\sin { x } +\cos { x } }

C.I=xsecxxsinx+cosxsecx(1+xtanx)xsinx+cosxdx\displaystyle C.\quad I= \frac { x\sec { x } }{ x\sin { x } +\cos { x } } -\int \frac { \sec { x } (1+x\tan { x) } }{ x\sin { x } +\cos { x } } dx

D.I=xsecxxsinx+cosx+secx(1+xtanx)xsinx+cosxdx\displaystyle D.\quad I= -\frac { x\sec { x } }{ x\sin { x } +\cos { x } } +\int \frac { \sec { x } (1+x\tan { x) } }{ x\sin { x } +\cos { x } } dx


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