QJEEM22Ma10a7: JEE Main 2022 Mathematics - Class 12 JEEM22Ma10a7 Solution

$\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {\left( {1 + {{\cos }^2}x} \right)\left( {{e^{\cos x}} + {e^{ - \cos x}}} \right)}}dx}$

Let $\cos x = t$

$\sin xdx = dt$

$= \int\limits_1^{ - 1} {{{ - {e^t}dt} \over {\left( {1 + {t^2}} \right)\left( {{e^t} + {e^{ - t}}} \right)}}}$

$I = \int\limits_{ - 1}^1 {{{{e^t}} \over {\left( {1 + {t^2}} \right)\left( {{e^t} + {e^{ - t}}} \right)}}dt}$ ...... (i)

$I = \int\limits_{ - 1}^1 {{{{e^{ - t}}} \over {\left( {1 + {t^2}} \right)\left( {{e^{ - t}} + {e^t}} \right)}}dt}$ .... (ii)

Adding (i) and (ii)

$2I = \int\limits_{ - 1}^1 {{{dt} \over {1 + {t^2}}}}$

$2I = \left. {{{\tan }^{ - t}}} \right|_{ - 1}^1$

$2I = {\pi \over 4} - \left( { - {\pi \over 4}} \right)$

$2I = {\pi \over 2}$

$I = {\pi \over 4}$