By using the properties of definite integrals, evaluate the integrals `int0pi/2(sinx-cosx)/(1+sinxcosx)dx`

Explanation

`"Let I "=int_(0)^(pi//2) (sinx -cos x)/(1+sin xcos x) dx......(1)`
`rArr " "I= int_(0)^(pi//2) (sin((pi)/(2)-x)-cos((pi)/(2)-x))/(1+sin((pi)/(2)-x)cos ((pi)/(2)-x))dx`
`[ :' int_(0)^(a) f(x) dx=int_(0)^(a) f(a-x)dx]`
`=int_(0)^(pi//2)(cos x-sin x)/(1+cos x sin x)dx" "....(2)`
Adding equations (1) and (2)
`2I =int_(0)^(pi//2) (sin x-cos x+ cos x-sin x)/(1+sin x cos x)dx`
`=0`
`rArr " "I=0`