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  4. The value of ∫ limits-π / 2π / 2 ( cos 2 x/1+3x) d x is

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Q. The value of $\int\limits_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x}{1+3^{x}} d x$ is

JEE MainJEE Main 2021Integrals

Solution:

$I =\int\limits_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x }{1+3^{ x }} dx$ (using king)
$I =\int\limits_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x }{1+3^{- x }} dx =\int\limits_{-\pi / 2}^{\pi / 2} \frac{3^{ x } \cos ^{2} x }{1+3^{ x }} dx$
$2 I =\int\limits_{-\pi / 2}^{\pi / 2} \frac{\left(1+3^{x}\right) \cos ^{2} x }{1+3^{x}} d x$
$=\int\limits_{-\pi / 2}^{\pi / 2} \cos ^{2} xdx =2 \int\limits_{0}^{\pi / 2} \cos ^{2} x\, dx$
$\Rightarrow I =\int\limits_{0}^{\pi / 2} \cos ^{2} x d x =\frac{\pi}{4}$