\rm\int_{ - (\pi)/(2) }^{ \frac{{\pi}}{2} } \frac{ \cos(x) }{1 + {cos}^( \tan(x) ) (x)} dx \\​

Question

`\rm\int_{ - (\pi)/(2) }^{ \frac{{\pi}}{2} } \frac{ \cos(x) }{1 + {cos}^( \tan(x) ) (x)} dx \\`​

Answer 1

Split the integral at
`x=0`. In the integral over
`\left[-\frac\pi2,0\right]`, substitute
`x\mapsto-x` to get

`\displaystyle \int_(-\pi/2)^0 (\cos(x))/(1 + \cos^(\tan(x))(x)) \, dx = \int_0^(\pi/2) (\cos(x))/(1 + \cos^(-\tan(x))(x)) \, dx`

Then we have

`\displaystyle \int_(-\pi/2)^(\pi/2) (\cos(x))/(1 + \cos^(\tan(x))(x)) \, dx \\\\ ~~~~~~~~ = \left\{\int_0^(\pi/2) + \int_(-\pi/2)^0\right\} (\cos(x))/(1 + \cos^(\tan(x))(x)) \, dx \\\\ ~~~~~~~~= \int_0^(\pi/2) \cos(x) \left(\frac1{1+\cos^(\tan(x))(x)} + \frac1{1 + \cos^(-\tan(x))(x)}\right) \, dx \\\\ ~~~~~~~~ = \int_0^(\pi/2) \cos(x) \left(\frac1{1 + \cos^(\tan(x))(x)} + (\cos^(\tan(x))(x))/(1 + \cos^(\tan(x))(x))\right) \, dx \\\\ ~~~~~~~~ = \int_0^(\pi/2) \cos(x) \, dx = \boxed{1}`