If `I_I=int_0^(pi//2)cos(sinx)dx ,I_2=int_0^(pi/2)sin(cosx)dx \ ,a n d \ I_3=int_0^(pi/2)cosx dx ,` then find the order in which the values `I_1,I_2,I_3,` exist.

Answer

`I_(1)gtI_(3)gtI_(2)`

Explanation

We know that `cosx` is decreasing function in `(0,pi//2)`.
Also `xgtsinx` for `xepsilon(0,pi//2)`
Thus, `cosx lt cos (sinx)`
Further, `xgtsinx` and `cosxepsilon(0,1)` for `xepsilon(0,pi//2)`
`:.cosxgtsin(cosx)`
Thus, `sin (cosx)ltcosxltcos(sinx)`
Hence `int_(0)^(pi//2)sin(cosx)dxlt int_(0)^(pi//2)cosxdx lt int_(0)^(pi//2) cos(sinx)dx`
`impliesI_(2)ltI_(3)ltI_(1)`