$K:=\int_{-\pi/2}^{\pi/2} {1+\sqrt{\cos(x)}}dx$ integral $K.M.N$ Who is the biggest and who is the smallest?

Question

Set $$M:=\int_{-\pi/2}^{\pi/2}\frac{((1+x)^2)}{(1+x^2)}dx$$ $$N:=\int_{-\pi/2}^{\pi/2}\frac{(1+x)}{(e^{x})}dx$$ $$K:=\int_{-\pi/2}^{\pi/2} {1+\sqrt{\cos(x)}}dx$$ integral $K.M.N$ Who is the biggest and who is the smallest?

Answer 1

It is easy to see that:$$M=\pi,N<\pi,K>\pi.$$ Please note that: $\frac{((1+x)^2)}{(1+x^2)}=1+\frac{2x}{1+x^2},$ $\frac{1+x}{e^x}<1$, $1+\sqrt{\cos(x)}>1$.