`IfI_1=int_0^(pi/2)(cos^2x)/(1+cos^2x)dx ,I_2=int_0^(pi/2)(sin^2x)/(1+sin^2x)dx` `I_3=int_0^(pi/2)(1+2cos^2xsin^2x)/(4+2cos^2xsin^2x)dx ,t h e n` `I_1=I_2> I_3` (b) `I_3> I_1=I_2` `I_1=I_2=I_3` (d) none of these

Options

A. `I_(1)=I_(2)gtI_(3)`
B. `I_(3)gtI_(1)=I_(2)`
C. `I_(1)=I_(2)=I_(3)`
D. none of these

Answer

C

Explanation

`I_(1)int_(0)^(pi//2)(cos^(2)x)/(1+cos^(2)x)dx`
`=int_(0)^(pi//2)(cos^(2)(pi//2-x))/(1+cos^(2)(pi//2-x))dx`
`=int_(0)^(pi//2) (sin^(2)x)/(1+sin^(2)x)dx=I_(2)`
Also `I_(1)+I_(2)=int_(0)^(pi//2)((sin^(2)x)/(1+sin^(2)x)+(cos^(2)x)/(1+cos^(2)x))dx`
`=int_(0)^(pi//2)(sin^(2)x+sin^(2)xcos^(2)x+cos^(2)x+sin^(2)xcos^(2)x)/(1+sin^(2)x+cos^(2)x+sin^(2)xcos^(2)x)`
`=int(1+2sin^(2)xcos^(2)x)/(2+sin^(2)x cos^(2)x)dx=2I_(3)`
`:. 2I_(1)=2I_(3)` or `I_(1)=I_(3)` or `I_(1)=I_(2)=I_(3)`