If `I_(1)= int_(1)^(sin theta) (x)/(1+x^(2)) dx and I_(2) int_(1)^("cosec" theta) (1)/(x(x^(2+1)))dx` then the value of `|{:(I_(1),I_(1)^(2),I_(2)),(e^(I_(1)+I_(2)),I_(2)^(2),-1),(1,I_(1)^(2)+I_(2)^(2),-1):}|` , is

Options

A. `sin theta`
B. `"cosec" theta`
C. 0
D. 1

Answer

C

Explanation

We have,
`I_(2)= underset(1)overset("cosec" theta)int (x)/(x(x^(2)+1))dx=-underset(1)overset(sin theta)int (t)/(1+t^(2))dt, "where"t(1)/(x)`
`rArr I_(2)=-I_(1)`
`:. |{:(I_(1),I_(1)^(2),I_(2)),(e^(I_(1)+I_(2)),I_(2)^(2),-1),(1,I_(1)^(2)+I_(2)^(2),-1):}|=|{:(I_(1),I_(1)^(2),-I_(1)),(e^(0),I_(1)^(2),-1),(1,2I_(1)^(2),-1):}|`
`|{:(I_(1),I_(1)^(2),I_(1)),(1,I_(1)^(2),-1),(0,I_(1)^(2),0):}| "Applying" R_(3) in R_(3)-R_(2)]`
`=-I_(1)^(2)(-I_(1)+I_(1))=0 " "` [ Expanding along `R_(3)`]