$\underset{0}{\overset{\pi /2}{\int }}\frac{\sec x}{\sec x+\cos ecx}dx=$

$I=\underset{0}{\overset{\pi /2}{\int }}\frac{\sec x}{\sec x+\cos ecx}dx$

$=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\sec \left(\frac{\pi }{2}-x\right)}{\sec \left(\frac{\pi }{2}-x\right)+\cos ec\left(\frac{\pi }{2}-x\right)}dx$

$\left(\because \underset{0}{\overset{a}{\int }}f\left(x\right)dx=\underset{0}{\overset{a}{\int }}f(a-x)dx\right)$

$I=\underset{0}{\overset{\pi /2}{\int }}\left(\frac{\cos ecx}{\cos ecx+\sec x}\right)dx$

$I+I=\underset{0}{\overset{\pi /2}{\int }}\left(\frac{\sec x}{\sec x+\cos ecx}+\frac{\cos ecx}{\cos ecx+\sec x}\right) dx$

$=\underset{0}{\overset{\pi /2}{\int }}\frac{\sec x+\cos ecx}{\sec x+\cos ecx} dx$

$=\underset{0}{\overset{\pi /2}{\int }}1dx={\left(x\right)}_0^{\pi /2}$

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

$I=\frac{\pi }{4}$