Scary Integration!

Calculus Level 5

\[\large \displaystyle \int \frac { { x }^{ 2 } }{ ({ x\sin { x } +\cos { x } ) }^{ 2 } } dx \]

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

\(\displaystyle A.\quad I= \frac { \sin { x } +x\cos { x } }{ x\sin { x } -\cos { x } } \)

\(\displaystyle B.\quad I= \frac { \sin { x } -x\cos { x } }{ x\sin { x } +\cos { x } } \)

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

\(\displaystyle D.\quad I= -\frac { x\sec { x } }{ x\sin { x } +\cos { x } } +\int \frac { \sec { x } (1+x\tan { x) } }{ x\sin { x } +\cos { x } } dx\)


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