QJEEM19Ma66d3: JEE Main 2019 Mathematics - Class 12 JEEM19Ma66d3 Solution: If = m( + n), then m.n is equal to

Question

Verified Solution • 2025–26 Session moderate

If

\int\limits_0^{{\pi \over 2}} {{{\cot x} \over {\cot x + \cos ecx}}} dx

= m(

\pi

+ n), then m.n is equal to

Accepted Answer

\int\limits_0^{\pi /2} {{{\cot x} \over {\cot x + {\mathop{\rm cosec}\nolimits} \,x}}dx = } \int\limits_0^{\pi /2} {{{\cos x} \over {\cos x + 1}}dx}

\Rightarrow \int\limits_0^{\pi /2} {{{2{{\cos }^2}{x \over 2} - 1} \over {2{{\cos }^2}{x \over 2}}}dx = } \int\limits_0^{\pi /2} {\left( {1 - {1 \over 2}{{\sec }^2}{x \over 2}} \right)dx}

\Rightarrow \left( {x - \tan {x \over 2}} \right)_0^{\pi /2} = {\pi \over 2} - 1 = {1 \over 2}\left( {\pi - 2} \right)

mn =

{1 \over 2}x - 2 = - 1

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