`I_1=int_0^(pi/2)(sinx-cosx)/(1+sinxcosx)dx ,I_2=int_0^(2pi)cos^6xdx ,I_3=int_(pi/2)^(pi/2)sin^3xdx ,I_4=int_0^1 1n(1/x-1)dxdotT h e n` `I_2=I_3=I_4=0,I_1!=0` `I_1=I_2=I_3=0,I_4!=0` `I_1=I_2=I_3=0,I_4!=0` `I_1=I_4=I_3=0,I_2!=0`

Options

A. `I_(2)=I_(3)=I_(4)=0,I_(1)!=0`
B. `I_(1)=I_(2)=I_(3)=0,I_(4)!=0`
C. `I_(1)=I_(3)=I_(4)=0,I_(2)!=0`
D. `I_(1)=I_(2)=I_(3)=0,I_(4)!=0`

Answer

C

Explanation

`I_(1)=int_(0)^(pi//2)(sinx_cosx)/(1+sinx cos) dx`
`=int_(0)^(pi//2) ("sin"((pi)/2-x)-"cos"((pi)/2-x))/(1+"sin"((pi)/2-x)"cos"((pi)/2-x))`
`=int_(0)^(pi//2)(cosx-sinx)/(1+sinx cosx)dx=-I_(1)`
or `I_(1)=0`
`I_(3)=0` as `sin^(3)x` is odd
`I_(4)=int_(0)^(1)In((1-x)/x)dx`
`=int_(0)^(1)In((1-(1-x))/(1-x))dx`
`=int_(0)^(1)In x/(1-x)dx=-I_(4)`
or `I_(4)=0`
`I_(2)=int_(0)^(2pi)cos^(6)x dx=2int_(0)^(pi)cos^(6)x dx!=0`