Evaluate: ∫_-(π)/2^((π)/2) (cos^2 x)/(2^x +1) d x

Let, $\displaystyle I=\int_{-\pi / 2}^{\pi / 2} \dfrac{\cos^{2} x}{2^{x}+1} dx \quad \ldots(1)$

Using property $\displaystyle \int_{a}^{b} f(x) dx=\int_{a}^{b} f(a+b-x) dx$, we get

$\displaystyle I=\int_{-\pi / 2}^{\pi / 2} \dfrac{2^{x} \cos^{2} x}{2^{x}+1} dx \quad \ldots(2)$

On adding equations ($\displaystyle 1$) and ($\displaystyle 2$), we get

$\displaystyle 2 I=\int_{-\pi / 2}^{\pi / 2} \cos^{2} x\, dx=2 \int_{0}^{\pi / 2} \cos^{2} x\, dx \Rightarrow I=\int_{0}^{\pi / 2} \cos^{2} x\, dx \quad \ldots(3)$

Getting $\displaystyle I=\int_{0}^{\pi / 2} \sin^{2} x\, dx \quad \ldots(4)$

On adding equations ($\displaystyle 3$) and ($\displaystyle 4$), we get

$\displaystyle 2 I=\int_{0}^{\pi / 2} 1\, dx \Rightarrow I=\dfrac{\pi}{4}$