QJEEM20Maf083: JEE Main 2020 Mathematics - Class 12 JEEM20Maf083 Solution: The integral is equal to (where C is a constant of integration):

Question

Verified Solution • 2025–26 Session moderate

The integral

\int {{{\left( {{x \over {x\sin x + \cos x}}} \right)}^2}dx}

is equal to
(where C is a constant of integration):

Accepted Answer

{\int {\left( {{x \over {x\sin x + \cos x}}} \right)} ^2}dx

= \int {\left( {{x \over {\cos x}}} \right).{{x\cos x\,dx} \over {{{(x\sin x + \cos x)}^2}}}}

=

{x \over {\cos x}}\left( { - {1 \over {x\sin x + \cos x}}} \right) + \int {\left( {{{\cos x + x\sin x} \over {{{\cos }^2}x}}} \right)} \left( {{1 \over {x\sin x + \cos x}}} \right)dx

= - {{x\sec x} \over {x\sin x + \cos x}} + \int {{{\sec }^2}x\,dx}

= - {{x\sec x} \over {x\sin x + \cos x}} + \tan x + C

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